The On-Line Encyclopedia of Integer Sequences, Recent Additions

This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

It shows the most recently added sequences in reverse chronological order.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    demo1.html:         Demos 
    Sindx.html:         Index 
    WebCam.html:        WebCam
    Submit.html:        Contribute new sequence or comment 
    eishelp1.html:      Internal format 
    eishelp2.html:      Beautified format
    transforms.html:    Transforms 
    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/
    

The WebCam at www.research.att.com/~njas/sequences/WebCam.html
is another way to browse the recent additions.

[If the database has just been resorted into lexicographic order,
the present file will be empty, but the WebCam will still work.]

(start)

%I A168356
%S A168356 1,0,1,8,3,3,3,3,0,5,4,1,0,4,7,0,6,1,2,2,0,2,3,1,5,0,6,4,7,
%T A168356 2,9,3,8,6,4,7,5,6,8,1,8,3,0,6,6,2,2,6,5,4,8,3,8,7,7,5,2,
%U A168356 0,5,1,5,3,6,2,4,8,6,2,3,6,2,2,2,4,6,6,6,9,1,1,3,6,2,2,5
%V A168356 1,0,1,8,-3,-3,3,-3,0,5,4,-1,0,-4,-7,0,6,1,-2,-2,0,2,-3,-1,5,0,-6,4,7,
%W A168356 -2,-9,-3,8,6,-4,-7,5,6,-8,-1,8,-3,0,6,-6,-2,2,-6,-5,4,8,-3,-8,7,7,-5,2,
%X A168356 0,-5,1,5,-3,-6,-2,4,8,-6,-2,3,-6,2,2,2,4,-6,-6,6,9,1,-1,-3,-6,2,-2,-5
%N A168356 A000796(n-2) - A000796(n)
%C A168356 Difference between digit n of pi and digit n+2.
%Y A168356 Cf. A000796
%K A168356 base,sign,new
%O A168356 1,4
%A A168356 Mark Lessel (mark(AT)lessel.us), Nov 23 2009

%I A168353
%S A168353 1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6,6,6,3,6,3,6,6,6,
%T A168353 3,6,6,6,6,6,3,3,6,6,3,3,3,3,3,3,6,6,6,6,6,6,6,3,6,6,6,6,3,3,6,3,3,6,6,
%U A168353 3,6,6,3,6,3,6,6,6,6,6,6,6,3,6,3,3,6,6,6,6,6,6,6,3,6,6,6,6,6,6,3,6,6,6
%N A168353 Number of distinct transpositions of digits of prime(n).
%Y A168353 Cf. A000040.
%K A168353 nonn,new
%O A168353 1,6
%A A168353 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 23 2009

%I A168355
%S A168355 165,195,231,255,273,455,561,645,903,1023,1105,1533,2015,2193,2289,2409,
%T A168355 2553,2829,3171,3219,3435,3855,3999,4161,4433,4953,5285,5397,5621,5709,
%U A168355 6141,6307,6643,7163,7239,7511,8481,9417,9705,10245,11805,12093,12291
%N A168355 Products of 3 distinct primes whose binary expansion is palindromic.
%e A168355 165=3*5*11={1,0,1,0,0,1,0,1} 195=3*5*13={1,1,0,0,0,0,1,1}
%t A168355 f1[n_]:=Reverse[IntegerDigits[n,2]]==IntegerDigits[n,2]; f2[n_]:=Last/@FactorInteger[n]=={1,1,1} lst={};Do[If[f1[n]&&f2[n],AppendTo[lst,n]],{n,8!}];lst
%K A168355 nonn,new
%O A168355 1,1
%A A168355 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 23 2009

%I A168267
%S A168267 1,2,4,6,8,12,24,36,48,60,72,96,120,180,240,360,480,720,840,1080,1260,
%T A168267 1440,1680,2160,2520,3360,4320,5040,7560,10080,15120,20160,25200,27720,
%U A168267 30240,40320,45360,50400,55440,75600,83160,110880,151200,166320,221760
%N A168267 Range of values of A168266.
%Y A168267 Includes A002182. Subsequence of A140999.
%Y A168267 Cf. A168264, A168265.
%K A168267 nonn,new
%O A168267 1,2
%A A168267 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168354
%S A168354 15,21,33,51,65,85,93,119,129,219,341,365,381,403,427,471,511,633,717,
%T A168354 771,843,951,1057,1137,1241,1273,1285,1317,1397,1501,1651,1707,1799,
%U A168354 1967,2047,2049,2661,2973,3579,3687,3831,4097,4321,4369,4529,4593,4681
%N A168354 Products of two distinct primes whose binary expansion is palindromic.
%e A168354 15=3*5={1,1,1,1} 21=3*7={1,0,1,0,1} 33=3*11={1,0,0,0,0,1}
%t A168354 f1[n_]:=Reverse[IntegerDigits[n,2]]==IntegerDigits[n,2]; f2[n_]:=Last/@FactorInteger[n]=={1,1} lst={};Do[If[f1[n]&&f2[n],AppendTo[lst,n]],{n,8!}];lst
%K A168354 nonn,new
%O A168354 1,1
%A A168354 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 23 2009

%I A168351
%S A168351 1,3,7,13,23,35,51,69,91,119,149,185,225,267,313,365,423,483,549,619,
%T A168351 691,769,851,939,1035,1135,1237,1343,1451,1563,1689,1819,1955,2093,2241,
%U A168351 2391,2547,2709,2875,3047,3225,3405,3595,3787,3983,4181,4391,4613,4839
%N A168351 Sum of first n primes minus n.
%F A168351 a(n)=A007504(n)-A000027(n)=(Sum{x=1..n}prime(x))-n.
%e A168351 a(1)=2-1=1, a(2)=2+3-2=3, a(3)=2+3+5-3=7, a(4)=2+3+5+7-4=13.
%K A168351 nonn,new
%O A168351 1,2
%A A168351 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 23 2009

%I A168266
%S A168266 1,1,2,1,2,4,1,2,4,6,1,2,4,6,8,1,2,4,6,8,12,1,2,4,6,8,12,1,2,4,6,8,12,1,
%T A168266 2,4,6,8,12,1,2,4,6,8,12,24,1,2,4,6,8,12,24,1,2,4,6,8,12,24,36,1,2,4,6,
%U A168266 8,12,24,36,1,2,4,6,8,12,24,36,1,2,4,6,8,12,24,36,1,2,4,6,8,12,24,36,48
%N A168266 A003557(A168264(n)).
%F A168266 Also A111701(A168264(n)).
%Y A168266 For range of values, see A168267.
%K A168266 nonn,new
%O A168266 1,3
%A A168266 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168350
%S A168350 0,1,4,9,16,24,33,43,54,67,81,96,112,129,148,168,189,211,235,260,286,
%T A168350 313,341,370,401,433,466,500,535,571,609,648,688,729,772,816,861,907,
%U A168350 955,1004,1054,1105,1157,1211,1266,1322,1379,1437,1496,1557,1619,1682
%N A168350 Sum of first n non-single or nonisolated numbers.
%F A168350 a(n)=sum{x=1..n}a(x).
%e A168350 a(1)=0, a(2)=0+1=1, a(3)=0+1+3=4.
%Y A168350 Cf. A168707.
%K A168350 nonn,new
%O A168350 1,3
%A A168350 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 23 2009

%I A168265
%S A168265 1,1,2,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,11,
%T A168265 12,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
%U A168265 13,14,15,16,17,18,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20
%N A168265 A003557(A060735(n)).
%C A168265 A060735(n) belongs to A168264 if and only if a(n) belongs to A168267.
%F A168265 Integers 1 to A006093(1) inclusive, followed by integers 1 to A006093(2) inclusive, etc.
%F A168265 Also A111701(A060735(n)).
%K A168265 easy,nonn,new
%O A168265 1,3
%A A168265 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168352
%S A168352 255255,285285,345345,373065,435435,440895,451605,465465,504735,533715,
%T A168352 555555,569415,596505,608685,615615,636405,645645,672945,680295,692835,
%U A168352 705705,719355,726495,752115,770385,780045,795795,803985,805035,811965
%N A168352 Odd numbers with exactly 6 distinct prime factors.
%e A168352 255255=3*5*7*11*13*17 285285=3*5*7*11*13*19 345345=3*5*7*11*13*23 435435=3*5*7*11*13*29
%t A168352 f[n_]:=Last/@FactorInteger[n]=={1,1,1,1,1,1}&&FactorInteger[n][[1,1]]>2; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6*9!}];lst
%K A168352 nonn,new
%O A168352 1,1
%A A168352 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 23 2009

%I A168264
%S A168264 1,2,4,6,12,24,30,60,120,180,210,420,840,1260,1680,2310,4620,9240,13860,
%T A168264 18480,27720,30030,60060,120120,180180,240240,360360,510510,1021020,
%U A168264 2042040,3063060,4084080,6126120,9699690,19399380,38798760,58198140
%N A168264 For all sufficiently high values of k, d(n^k) > d(m^k) for all m < n. (Let k, m, and n represent positive integers only.)
%C A168264 d(n) is the number of divisors of n (A000005(n)).
%H A168264 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial Calculator</a>
%H A168264 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%H A168264 G. Xiao, WIMS server, <a href="http://wims.unice.fr/wims/wims.cgi?module=tool/algebra/factor.en">Factoris</a> (both expands and factors polynomials)
%F A168264 If the canonical factorization of n into prime powers is Product p^e(p), then the formula for the number of divisors of the kth power of n is Product_p (ek + 1). (See also A146289, A146290.)
%F A168264 For two positive integers m and n with different prime signatures, let j be the largest exponent of k for which m and n have different coefficients, after the above formula for each integer is expanded as a polynomial. Let m_j and n_j denote the corresponding coefficients. d(n^k) > d(m^k) for all sufficiently high values of k if and only if n_j > m_j.
%e A168264 Since the exponents in 1680's prime factorization are (4,1,1,1), the kth power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following:
%e A168264 a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4.
%e A168264 b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1).
%e A168264 c) The k^3 coefficient for 1680 is higher than for 1260.
%e A168264 So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680.
%Y A168264 Subsequence of A025487, A060735, A116998. Includes A002110, A168262, A168263.
%Y A168264 See also A168265, A168266, A168267.
%K A168264 nonn,new
%O A168264 1,2
%A A168264 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168348
%S A168348 41,53,67,113,127,139,151,191,269,307,353,409,461,491,547,619,701,829,
%T A168348 919,1031,1063,1193,1231,1249,1289,1607,1667,1721,1759,2089,2131,2179,
%U A168348 2281,2381,2467,2609,2647,2851,2861,2953,3221,3331,3361,3391,3407,3571
%N A168348 Primes which are the sum of two consecutive single or isolated numbers.
%e A168348 a(1)=A168706(5)+A168706(6)=18+23=41.
%Y A168348 Cf. A000040, A168706.
%K A168348 nonn,new
%O A168348 1,1
%A A168348 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 23 2009

%I A168349
%S A168349 174,253,292,368,371,375,2212,2219,16815,16816,78133,78134
%N A168349 Numbers of the form a^b * c^d where a, b, c and d are the first 4 primes.
%C A168349 (1) There are 4! = 24 permutations of 4 elements, because of commutativity of multiplication sequences has 12 different terms
%C A168349 (2) Note that all terms are composite
%D A168349 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D A168349 Friedhelm Padberg, Elementare Zahlentheorie, Spektrum Akademischer Verlag, 2. Auflage 1991
%e A168349 (1) 174 = 2 x 3 x 29 = 5^3 + 7^2
%e A168349 (2) 253 = 11 x 23 = 2^7 + 5^3
%e A168349 (3) 292 = 2 ^ 2 x 73 = 3^5 + 7^2
%e A168349 (4) 368 = 2^4 x 23 = 5^2 + 7^3
%e A168349 (5) 371 = 7 x 53 = 2^7 + 3^5
%e A168349 (6) 375 = 3 x 5 ^ 3 = 2^5 + 7^3
%e A168349 (7) 2212 = 2 ^ 2 x 7 x 79 = 3^7 + 5^2
%e A168349 (8) 2219 = 7 x 317 = 2^5 + 3^7
%e A168349 (9) 16815 = 3 x 5 x 19 x 59 = 2^3 + 7^5
%e A168349 (10) 16816 = 2 ^ 4 x 1051 = 3^2 + 7^5
%e A168349 (11) 78133 = 11 x 7103 = 2^3 + 5^7
%e A168349 (12) 78134 = 2 x 7 x 5581 = 3^2 + 5^7
%K A168349 fini,nonn,new
%O A168349 1,1
%A A168349 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 23 2009

%I A168263
%S A168263 1,2,4,6,12,24,60,120,180,840,1260,1680,27720
%N A168263 For any m < n, and for all values of k, d(n^k) > d(m^k). (Let k, m, and n represent positive integers only.)
%C A168263 d(n) is the number of divisors of n (A000005(n)).
%C A168263 All members must be highly composite numbers (A002182) with at least as many distinct prime factors as any smaller positive integer (A116998). (See Formula and Example sections.) It turns out that these two conditions are jointly sufficient.
%C A168263 Ramanujan proved that a) for any prime p, there exist a finite number of highly composite numbers with p as its largest prime factor; and b) in the canonical prime factorization of a highly composite number with largest prime factor p, the exponents for all primes > p are never smaller than they are in the factorization of A003418(p). (See formula 54 of the Ramanujan paper.)
%C A168263 It follows that, if the intersection of A003418 and A116998 is finite, so is the intersection of A002182 and A116998. For proof that the former intersection is finite, see A168262.
%D A168263 S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
%H A168263 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial Calculator</a>
%H A168263 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page15.htm">Highly Composite Numbers (p. 15)</a> (note especially pp. 11-15)
%H A168263 G. Xiao, WIMS server, <a href="http://wims.unice.fr/wims/wims.cgi?module=tool/algebra/factor.en">Factoris</a> (both expands and factors polynomials)
%F A168263 If the canonical factorization of n into prime powers is Product p^e(p), then the formula for the number of divisors of the kth power of n is Product_p (ek + 1). (See also A146289, A146290.)
%F A168263 For two positive integers m and n with different prime signatures, let j be the largest exponent of k for which m and n have different coefficients, after the above formula for each integer is expanded as a polynomial. Let m_j and n_j denote the corresponding coefficients. d(n^k) > d(m^k) for all sufficiently high values of k if and only if n_j > m_j.
%e A168263 1) 1680 has more divisors than any smaller positive integer; thus for all m < n, d(1680^1) > d(m^1).
%e A168263 2) Since the exponents in 1680's prime factorization are (4,1,1,1), the kth power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following:
%e A168263 a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4.
%e A168263 b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1).
%e A168263 c) The k^3 coefficient for 1680 is higher than for 1260.
%e A168263 So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680.
%e A168263 3) Careful comparison of 1680's "power formula" with the analogous formulas for smaller members of A025487 shows that no intermediate value of k can exist for which d(m^k) >= d(1680^k) if m < 1680.
%Y A168263 Intersection of A002182 and A116998. Also, intersection of A002182 and A060735, and of A002182 and A168264. (A168264 is a subsequence of A060735, which is a subsequence of A116998.)
%K A168263 fini,full,nonn,new
%O A168263 1,2
%A A168263 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168347
%S A168347 1,1,2,1,1,7,7,9,1,18,39,68,55,10,1,1,41,181,381,691,395,215,15,1,88,
%T A168347 733,2048,5378,6512,5026,2816,381,56,1,1,183,2703,10921,34826,71590,
%U A168347 78590,76146,34853,11123,1603,21,1,374,9355,56668,211865,627434,1000219
%N A168347 Coefficients of the expansion of:p(x,t)=(1 - x)/((1 - x*Exp[t*(1 - x)])*(1 - x*Exp[t*(1 + x)]))
%C A168347 Row sums are:
%C A168347 {1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600,...}
%F A168347 p(x,t)=(1 - x)/((1 - x*Exp[t*(1 - x)])*(1 - x*Exp[t*(1 + x)]))
%e A168347 {1},
%e A168347 {1, 2, 1},
%e A168347 {1, 7, 7, 9},
%e A168347 {1, 18, 39, 68, 55, 10, 1},
%e A168347 {1, 41, 181, 381, 691, 395, 215, 15},
%e A168347 {1, 88, 733, 2048, 5378, 6512, 5026, 2816, 381, 56, 1},
%e A168347 {1, 183, 2703, 10921, 34826, 71590, 78590, 76146, 34853, 11123, 1603, 21},
%e A168347 {1, 374, 9355, 56668, 211865, 627434, 1000219, 1284488, 1109403, 573546, 245337, 37724, 4299, 246, 1},
%e A168347 {1, 757, 31009, 282445, 1275829, 4828105, 10948261, 17566417, 22506403, 17850207, 11567907, 4707207, 1126119, 197667, 8919, 27},
%e A168347 {1, 1524, 99777, 1350960, 7699308, 34659024, 106620396, 214473168, 357213030, 407969912, 351650086, 236212432, 99526828, 33716944, 6116140, 587824, 47233, 1012, 1}
%t A168347 p[t_] = (1 - x)/((1 - x*Exp[t*(1 - x)])*(1 - x*Exp[t*(1 + x)]))
%t A168347 a = Table[ CoefficientList[FullSimplify[ExpandAll[n!*(( 1 - x)^(n + 1)/(2*x))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 1, 10}]
%t A168347 Flatten[a]
%K A168347 nonn,uned,new
%O A168347 1,3
%A A168347 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 23 2009

%I A168344
%S A168344 1,1,3,15,99,773,6743,63591,635307,6634599,71759983,798563065,
%T A168344 9098321475,105733563393,1249676348391,14986826364311,182027688352427,
%U A168344 2235713532561779,27732857308708571,347064951865766607
%N A168344 G.f. A(x) satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = g.f. of A006664, which is the number of irreducible systems of meanders.
%F A168344 G.f.: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A001246, which is the squares of Catalan numbers.
%F A168344 G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A001246.
%F A168344 G.f.: A(x) = (1/x)*Series_Reversion(x/F(x)) where G(x) = g.f. of A006664.
%e A168344 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 773*x^5 + 6743*x^6 +...
%e A168344 A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A001246:
%e A168344 F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +...
%e A168344 A(x) satisfies: A(x/G(x)) = G(x) = g.f. of A006664:
%e A168344 G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...
%o A168344 (PARI) {a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)^2));polcoeff(x/serreverse(x*Ser(C_2)),n)}
%Y A168344 Cf. A006664, A001246, A000108.
%K A168344 nonn,new
%O A168344 0,3
%A A168344 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A168343
%S A168343 1,2,3,8,13,17,23,29,33,37,42,48,54,58,64,67,72,84,89,93,106,109,115,
%T A168343 126,132,137,140,145,151,162,167,179,190,196,198,204,214,219,224,230,
%U A168343 236,240,250,263,267,271,284,289,299,303,308,315,320,325,328,333,340
%N A168343 nth single or isolated number minus n.
%F A168343 a(n)=A167706(n)-A000027(n)=A167706(n)-n.
%K A168343 nonn,new
%O A168343 1,2
%A A168343 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 23 2009

%I A168262
%S A168262 1,2,6,12,60,420,840,27720,360360,5354228880
%N A168262 Intersection of A003418 and A116998.
%C A168262 If, for some prime p, A045948(p) > p^2, then all members of the sequence are less than A003418(p). (Let p_(n) be a prime for which the inequality is satisfied, and let p_(n+1) be the smallest prime > (p_(n))^2. No number smaller than A003418(p_(n+1)) can belong to this sequence. However, for any p_(n) that satisfies the inequality, so does p_(n+1), leading to an endless cycle.) This inequality is first satisfied at p=53, as A045948(53)=5040 > 53^2=2809.
%C A168262 Proof: It follows from the definitions of p_(n) and p_(n+1), and from Bertrand's Postulate, that 2(A045948(p_(n))) > 2((p_(n))^2) > p_(n+1). Therefore 2((A045948(p_(n)))^2 > (p_(n+1))^2.
%C A168262 Since any prime that divides A003418(p_(n)) divides A003418(p_(n+1)) at least twice as often, A045948(p_(n+1)) cannot be less than the product of (A045948(p_n))^2 and A034386(p_(n)). (The latter term greatly exceeds 2 for any actual p_(n).)
%C A168262 Therefore A045948(p_(n+1)) > 2((A045948(p_n))^2 > (p_(n+1))^2, and p_(n+1) satisfies the inequality, implying that no number smaller than A003418(p_(n+2)) can belong to this sequence.
%H A168262 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%Y A168262 Also intersection of A003418 and A060735, and of A003418 and A168264. (A168264 is a subsequence of A060735, which is a subsequence of A116998.)
%Y A168262 See also A001221, A168263.
%K A168262 fini,full,nonn,new
%O A168262 1,2
%A A168262 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 23 2009

%I A168342
%S A168342 2,4,12,10,8,6,20,18,16,14,38,36,34,32,30,28,26,24,22,56,54,52,50,48,46,
%T A168342 44,42,40,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,120,118,116,
%U A168342 114,112,110,108,106,104,102,100,98,96,94,92,90
%N A168342 Even elements via Janet periodic table of 120 (union of A138100 and A138101;see A168142).
%C A168342 A permutation of positive evens A087113=A005843(n+1). First 60 terms,by 1,1,4,4,9,9,16,16=A008794(n+2) . (In reference (1) of A168142=2,1,8,7,6,5, Janet mentions his first publication: Essai de schematisation de la structure des noyaux atomiques,mai 1927 (19 pages), not seen).
%K A168342 nonn,uned,new
%O A168342 1,1
%A A168342 Paul Curtz (bpcrtz(AT)free.fr), Nov 23 2009

%I A168341
%S A168341 0,1,3,4,9,6,16,8,25,36,11,49,13,64,15,81,100,18,121,20,144,22,169,24,
%T A168341 196,225,27,256,29,289,31,324,33,361,35,400,441,38,484,40,529,42,576,44,
%U A168341 625,46,676,48,729,784,51,841,53,900,55,961,57,1024,59,1089,61,1156,63
%N A168341 The lexicographically smallest injective sequence of nonnegative integers such that a(a(n)) is a square for all n>=0.
%C A168341 The term a(n) is either n+1 or a square. All the squares appear and they appear in increasing order. Every other term is a square, except when the index is a square, in which case, the corresponding term is also a square (which shifts the pattern). See FORMULA for a more precise statement.
%F A168341 To define a(n), let k = floor(sqrt(n)). Then a(n) = n+1 if n-k^2 is odd and ((n+k)/2)^2 if n-k^2 is even.
%F A168341 Note that k^2 is the largest square which is at most n.
%e A168341 For n=6, we have k=floor(sqrt(6))=2; since 6-2=4 is even, a(6)=((6+2)/2)^2=16.
%K A168341 nonn,new
%O A168341 0,3
%A A168341 Eric Angelini and Benoit Jubin (benoit_jubin(AT)yahoo.fr), Nov 23 2009

%I A168340
%S A168340 101111111111,111011111111,111111110111,111111111101
%N A168340 Near repunit emirps.
%Y A168340 Cf. A065074
%K A168340 more,nonn,uned,base,new
%O A168340 1,1
%A A168340 Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 23 2009

%I A168161
%S A168161 3,5,11,19,23,47,61
%N A168161 Primes p which are equal to the sum of the binary digits of
%C A168161 A subsequence of A168162.
%F A168161 A168161 = { prime p | p=A095375(pi(p)) } = { prime(n) |
%o A168161 (PARI) s=0; forprime(p=1,9999, if(p==s+=norml2(binary(p)),
%K A168161 fini,full,nonn,new
%O A168161 1,1
%A A168161 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009

%I A166899
%S A166899 1,3,25,111,456,2697,15961,86247,495781,3003738,17946798,107667969,
%T A166899 660458787,4081397547,25274724105,157744019799,991384251102,
%U A166899 6254115981009,39613066988527,252017709962526,1608980424431755
%N A166899 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k), n>=1.
%F A166899 Logarithmic derivative of A166898.
%e A166899 L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 +...
%e A166899 exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...+ A166898(n)*x^n +...
%o A166899 (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^4*n/(n-k))
%Y A166899 Cf. A166898, variants: A167539, A166895, A166897.
%K A166899 nonn,new
%O A166899 0,2
%A A166899 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A166898
%S A166898 1,1,2,10,38,137,646,3241,15623,79439,427562,2317396,12715372,71543343,
%T A166898 408543758,2353591560,13717994046,80827739181,480016288156,
%U A166898 2871701561720,17304832805996,104933348346951,639814473417775
%N A166898 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k] * x^n/n ), an integer series in x.
%F A166898 G.f.: exp( Sum_{n>=1} A166899(n)*x^n/n ) where A166899(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k).
%e A166898 G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...
%e A166898 log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 + 15961*x^7/7 +...+ A166899(n)*x^n/n +...
%o A166898 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)*x^m/m)+x*O(x^n)), n)}
%o A166898 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^4*m/(m-k))*x^m/m)+x*O(x^n)), n)}
%Y A166898 Cf. A166897, variants: A166894, A166898.
%K A166898 nonn,new
%O A166898 0,3
%A A166898 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A167539
%S A167539 1,3,7,15,36,87,211,519,1285,3198,7998,20079,50571,127725,323367,820407,
%T A167539 2085306,5309169,13537045,34561890,88347091,226079208,579110262,
%U A167539 1484766015,3809948461,9783998877,25143452881,64658016249,166375274790
%N A167539 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2*n/(n-k), n>=1.
%e A167539 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 36*x^5/5 + 87*x^6/6 +...
%e A167539 exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A004148(n+1)*x^n/n +...
%o A167539 (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^2*n/(n-k))
%Y A167539 Cf. A004148, variants: A166895, A166897, A166899.
%K A167539 nonn,new
%O A167539 1,2
%A A167539 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A166897
%S A166897 1,3,13,39,126,477,1765,6495,24709,95128,367368,1431453,5620343,
%T A166897 22170543,87858813,349708431,1397003136,5598513261,22502171771,
%U A166897 90681323364,366299212873,1482827487650,6014529069540,24439715146941
%N A166897 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.
%F A166897 Logarithmic derivative of A166896.
%e A166897 L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 +...
%e A166897 exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A166896(n)*x^n/n +...
%o A166897 (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^3*n/(n-k))
%Y A166897 Cf. A166897, variants: A167539, A166895, A166899.
%K A166897 nonn,new
%O A166897 1,2
%A A166897 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A166896
%S A166896 1,1,2,6,16,45,142,459,1508,5122,17787,62649,223971,811339,2970032,
%T A166896 10974150,40893393,153512844,580082454,2205046961,8427087958,
%U A166896 32362949488,124837337235,483508287359,1879669861074,7332469937755
%N A166896 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n/n ), an integer series in x.
%F A166896 G.f.: exp( Sum_{n>=1} A166897(n)*x^n/n ) where A166897(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k).
%e A166896 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...
%e A166896 log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 + 1765*x^7/7 +...+ A166897(n)*x^n/n +...
%o A166896 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)*x^m/m)+x*O(x^n)), n)}
%o A166896 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^3*m/(m-k))*x^m/m)+x*O(x^n)), n)}
%Y A166896 Cf. A166897, variants: A166894, A166898.
%K A166896 nonn,new
%O A166896 0,3
%A A166896 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A168339
%S A168339 8,13,18,21,20
%N A168339 a(n) is least number of squares needed to form n holes inside a rectangle
%e A168339 a(5)=20 because to create a rectangle with 5 holes inside, 20 squares are needed, as follows:
%e A168339 HHHHH
%e A168339 H H H
%e A168339 HH HH
%e A168339 H H H
%e A168339 HHHHH
%e A168339 But a(4)=21, because another square is needed to form 4 holes:
%e A168339 HHHHH
%e A168339 H H H
%e A168339 HHHHH
%e A168339 H H H
%e A168339 HHHHH
%K A168339 nice,nonn,more,new
%O A168339 1,1
%A A168339 Zhining Yang (northwolves(AT)163.com), Nov 23 2009
%E A168339 Edited by njas, Nov 23 2009

%I A168338
%S A168338 1,3,4,7,6,12,8,15,13,9,2,18,4,14,14,21,8,29,10,15,13,6,4,30,11,12,20,
%T A168338 26,10,26,4,24,8,14,18,41,8,20,16,27,5,29,5,14,28,12,8,44,17,19,16,21,6,
%U A168338 41,12,40,20,20,10,40,7,12,28,30,15,24,8,26,16,32,8,60,8,17,26,31,16,38
%N A168338 Sum of largest digit of divisors of n.
%C A168338 a(16)=21 because the divisors of 16 are [1, 2, 4, 8, 16] and 1+2+4+8+6 = 21.
%D A168338 J. Earls, "Black Hole 14," Mathematical Bliss, Pleroma Publications, 2009, pages 18-22. ASIN: B002ACVZ6O
%K A168338 base,easy,nonn,new
%O A168338 1,2
%A A168338 Jason Earls (zevi_35711(AT)yahoo.com), Nov 23 2009

%I A168337
%S A168337 1,8,8,15,15,22,22,29,29,36,36,43,43,50,50,57,57,64,64,71,71,78,78,85,
%T A168337 85,92,92,99,99,106,106,113,113,120,120,127,127,134,134,141,141,148,148,
%U A168337 155,155,162,162,169,169,176,176,183,183,190,190,197,197,204,204,211
%N A168337 a(n)=7*n-a(n-1)-5 (with a(1)=1)
%F A168337 a(n)=7*n-a(n-1)-5 (with a(1)=1)
%e A168337 For n=2, a(2)=7*2-1-5=8; n=3, a(3)=7*3-8-5=8; n=4, a(4)=7*4-8-5=15
%K A168337 nonn,new
%O A168337 1,2
%A A168337 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168336
%S A168336 5,5,12,12,19,19,26,26,33,33,40,40,47,47,54,54,61,61,68,68,75,75,82,82,
%T A168336 89,89,96,96,103,103,110,110,117,117,124,124,131,131,138,138,145,145,
%U A168336 152,152,159,159,166,166,173,173,180,180,187,187,194,194,201,201,208
%N A168336 a(n)=7*n-a(n-1)-4 (with a(1)=5)
%F A168336 a(n)=7*n-a(n-1)-4 (with a(1)=5)
%e A168336 For n=2, a(2)=7*2-5-4=5; n=3, a(3)=7*3-5-4=12; n=4, a(4)=7*4-12-4=12
%K A168336 nonn,new
%O A168336 1,1
%A A168336 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168335
%S A168335 0,1,0,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A168335 131072,262144,524288,1048576
%N A168335 Binomial transform of 0,1,-2,5,-8,13,-18,25,=A000982 signed.
%C A168335 Note A084633=1,0,2,4,8,16,=A131577 with 0,1 swapped. Like submitted A166954= 2,1,4,8,16,32,64,=2,A151821=A000079 with 1,2 swapped.
%F A168335 a(n)=0,(A034008=A084633 unsigned).
%K A168335 nonn,uned,new
%O A168335 0,4
%A A168335 Paul Curtz (bpcrtz(AT)free.fr), Nov 23 2009

%I A168334
%S A168334 1023456987896543201,1023458697968543201,1023459768679543201,
%T A168334 1023469587859643201
%N A168334 Pandigital palindromic primes.
%C A168334 Intersection of A050288 and A002385.
%Y A168334 A050288, A002385
%K A168334 more,nonn,uned,new
%O A168334 1,1
%A A168334 Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 23 2009

%I A168333
%S A168333 2,9,9,16,16,23,23,30,30,37,37,44,44,51,51,58,58,65,65,72,72,79,79,86,
%T A168333 86,93,93,100,100,107,107,114,114,121,121,128,128,135,135,142,142,149,
%U A168333 149,156,156,163,163,170,170,177,177,184,184,191,191,198,198,205,205
%N A168333 a(n)=7*n-a(n-1)-3 (with a(1)=2)
%F A168333 a(n)=7*n-a(n-1)-3 (with a(1)=2)
%e A168333 For n=2, a(2)=7*2-2-3=9; n=3, a(3)=7*3-9-3=9; n=4, a(4)=7*4-9-3=16
%K A168333 nonn,new
%O A168333 1,1
%A A168333 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168332
%S A168332 6,6,13,13,20,20,27,27,34,34,41,41,48,48,55,55,62,62,69,69,76,76,83,83,
%T A168332 90,90,97,97,104,104,111,111,118,118,125,125,132,132,139,139,146,146,
%U A168332 153,153,160,160,167,167,174,174,181,181,188,188,195,195,202,202,209
%N A168332 a(n)=7*n-a(n-1)-2 (with a(1)=6)
%F A168332 a(n)=7*n-a(n-1)-2 (with a(1)=6)
%e A168332 For n=2, a(2)=7*2-6-2=6; n=3, a(3)=7*3-6-2=13; n=4, a(4)=7*4-13-2=13
%K A168332 nonn,new
%O A168332 1,1
%A A168332 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168331
%S A168331 3,10,10,17,17,24,24,31,31,38,38,45,45,52,52,59,59,66,66,73,73,80,80,87,
%T A168331 87,94,94,101,101,108,108,115,115,122,122,129,129,136,136,143,143,150,
%U A168331 150,157,157,164,164,171,171,178,178,185,185,192,192,199,199,206,206
%N A168331 a(n)=7*n-a(n-1)-1 (with a(1)=3)
%F A168331 a(n)=7*n-a(n-1)-1 (with a(1)=3)
%e A168331 For n=2, a(2)=7*2-3-1=10; n=3, a(3)=7*3-10-1=10; n=4, a(4)=7*4-10-1=17
%K A168331 nonn,new
%O A168331 1,1
%A A168331 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168330
%S A168330 3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,
%T A168330 3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,
%U A168330 3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2
%V A168330 3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,
%W A168330 3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,
%X A168330 3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2
%N A168330 Period 2: repeat 3, -2.
%C A168330 Interleaving of A010701 and -A007395.
%C A168330 Binomial transform of 3 followed by a signed version of A020714.
%C A168330 Inverse binomial transform of 3 followed by A000079.
%C A168330 a(n+1)-a(n) = 5*(-1)^n.
%C A168330 A084964 without first two terms gives partial sums.
%F A168330 a(n) = (-5*(-1)^n+1)/2.
%F A168330 a(n) = -a(n-1)+1 for n > 1; a(1) = 3.
%F A168330 a(n) = a(n-2) for n > 2; a(1) = 3, a(2) = -2.
%F A168330 G.f.: x*(3-2*x)/((1-x)*(1+x)).
%o A168330 (MAGMA) &cat[ [3, -2]: n in [1..42] ];
%o A168330 [ n eq 1 select 3 else -Self(n-1)+1: n in [1..84] ];
%Y A168330 Cf. A168309 (repeat 4, -3), A010701 (all 3's sequence), A007395 (all 2's sequence), A010716 (all 5's sequence), A020714 (5*2^n), A000079 (powers of 2), A084964 (follow n+2 by n).
%K A168330 sign,new
%O A168330 1,1
%A A168330 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 23 2009

%I A168329
%S A168329 3,3,9,9,15,15,21,21,27,27,33,33,39,39,45,45,51,51,57,57,63,63,69,69,75,
%T A168329 75,81,81,87,87,93,93,99,99,105,105,111,111,117,117,123,123,129,129,135,
%U A168329 135,141,141,147,147,153,153,159,159,165,165,171,171,177,177,183,183
%N A168329 a(n)=6*n-a(n-1)-6 (with a(1)=3)
%e A168329 a(n)=6*n-a(n-1)-6 (with a(1)=3)
%p A168329 For n=2, a(2)=6*2-3-6=3; n=3, a(3)=6*3-3-6=9; n=4, a(4)=6*4-9-6=9
%K A168329 nonn,new
%O A168329 1,1
%A A168329 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168328
%S A168328 0,6,6,12,12,18,18,24,24,30,30,36,36,42,42,48,48,54,54,60,60,66,66,72,
%T A168328 72,78,78,84,84,90,90,96,96,102,102,108,108,114,114,120,120,126,126,132,
%U A168328 132,138,138,144,144,150,150,156,156,162,162,168,168,174,174,180,180
%N A168328 a(n)=6*n-a(n-1)-6 (with a(1)=0)
%F A168328 a(n)=6*n-a(n-1)-6 (with a(1)=0)
%e A168328 For n=2, a(2)=6*2-0-6=6; n=3, a(3)=6*3-6-6=6; n=4, a(4)=6*4-6-6=12
%K A168328 nonn,new
%O A168328 1,2
%A A168328 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A168327
%S A168327 11,127,12197,135937,159319,11092727,11295029,11860867,12685619,
%T A168327 14330747,14826809,15000211,15929741,16128487,18869743,19393931,
%U A168327 124137569,126198073,127818127,129503629,138958219,150243409,154439939
%N A168327 Primes of concatenated form p= "1 n^3"
%C A168327 (1) It is conjectured that sequence is infinite
%C A168327 (2) These are primes all with "leading" digit "1", they are concatenations of two cubic numbers: 1^3 and n^3, n is a natural
%D A168327 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A168327 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D A168327 Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996
%F A168327 If n^3 is a d-digit number and d no multiple of 3, then p=10^d+n^3, where n is odd and no multiple of 5
%e A168327 (1) 10^1+1^3=11=prime(5)=a(1)
%e A168327 (2) 10^2+3^3=127=prime(31)=a(2)
%e A168327 (3) 10^4+13^3=12197=prime(1458)=a(3)
%Y A168327 Cf. A000040 The prime numbers
%Y A168327 Cf. A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
%Y A168327 Cf. A167535 Concatenation of two square numbers which give a prime
%K A168327 nonn,new
%O A168327 1,1
%A A168327 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 23 2009

%I A168326
%S A168326 4,4,10,10,16,16,22,22,28,28,34,34,40,40,46,46,52,52,58,58,64,64,70,70,
%T A168326 76,76,82,82,88,88,94,94,100,100,106,106,112,112,118,118,124,124,130,
%U A168326 130,136,136,142,142,148,148,154,154,160,160,166,166,172,172,178,178
%N A168326 a(n)=6*n-a(n-1)-4 (with a(1)=4)
%F A168326 a(n)=6*n-a(n-1)-4 (with a(1)=4)
%e A168326 For n=2, a(2)=6*2-4-4=4; n=3, a(3)=6*3-4-4=10; n=4, a(4)=6*4-10-4=10
%K A168326 nonn,new
%O A168326 1,1
%A A168326 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009

%I A166895
%S A166895 1,3,7,39,366,5697,194881,16288695,2430565261,564615230758,
%T A166895 257227244037248,319346787227133873,832952161388710135215,
%U A166895 3382434539389226013260403,22966972221673234523620345857
%N A166895 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.
%F A166895 Logarithmic derivative of A166894.
%e A166895 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...
%e A166895 exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...+ A166894(n)*x^n +...
%o A166895 (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^(n-k)*n/(n-k))
%Y A166895 Cf. A166894.
%K A166895 nonn,new
%O A166895 1,2
%A A166895 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A166894
%S A166894 1,1,2,4,14,89,1050,28983,2066217,272159513,56735786726,23441305184736,
%T A166894 26635730598676118,64099902414443754551,241666593661232949435382,
%U A166894 1531373212165249576810266758,24642808245610936988728333582900
%N A166894 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^n * x^k] * x^n/n ), an integer series in x.
%F A166894 G.f.: exp( Sum_{n>=1} A166895(n)*x^n/n ) where A166895(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k).
%e A166894 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...
%e A166894 log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...+ A166895(n)*x^n/n +...
%o A166894 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^m*x^k)*x^m/m)+x*O(x^n)), n)}
%o A166894 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^(m-k)*m/(m-k))*x^m/m)+x*O(x^n)), n)}
%Y A166894 Cf. A166895.
%K A166894 nonn,new
%O A166894 0,3
%A A166894 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

%I A168325
%S A168325 3,11,59,67,83,103,107,131
%N A168325 Mangammal prime (cf. A123239) which are also the natural primes of k(i).
%Y A168325 Cf. A123239
%K A168325 nonn,new
%O A168325 1,1
%A A168325 A. K. Devaraj (dkandadai(AT)gmail.com), Nov 23 2009

%I A168324
%S A168324 0,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,3,1,3,2,2,1,4,1,2,1,3,1,6,1,1,2,2,2,
%T A168324 6,1,2,2,4,1,6,1,3,3,2,1,5,1,3,2,3,1,4,2,4,2,2,1,12,1,2,3,1,2,6,1,3,2,6,
%U A168324 1,10,1,2,3,3,2,6,1,5,1,2,1,12,2,2,2,4,1,12,2,3,2,2,2,6,1,3,6,1,6,1,4,6
%N A168324 Number of distinct transpositions of prime factors of n, where a(1)=0 and a(n=prime)=1.
%e A168324 a(18)=3 because 18=2*2*3=2*3*2=3*2*2; a(24)=4 because 24=2*2*2*3=2*2*3*2=2*3*2*2=3*2*2*2; a(26)=2 because 26=2*13=13*2; a(30)=6 because 30=2*3*5=2*5*3=3*2*5=3*5*2=5*2*3=5*3*2.
%Y A168324 Cf. A066882.
%K A168324 nonn,new
%O A168324 1,6
%A A168324 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168323
%S A168323 3,5,11,7,13,11,17,13,23,17,31,19,47,23,61,29,67,31,83,37,103,41,107,43,
%T A168323 113,67,127,83,137,97,139,101,149,103,157,107,167,109,173,127,179,137,
%U A168323 193,149,199,151,227,163,229,179,233,181,239,193,241,197,263,199,271
%N A168323 a(1)=3,a(2)=5; a(n+1)=smallest prime number > a(n-1) and Not equals to a(n) such that the sum of any three consecutive terms is a prime.
%t A168323 a=3;b=5;lst={a,b};Do[Do[If[PrimeQ[q]&&PrimeQ[a+b+q]&&q!=b,c=q;Break[]],{q,a+2,9!,2}];AppendTo[lst,c];a=b;b=c,{n,6!}];lst
%Y A168323 Cf. A062391, A168322
%K A168323 nonn,new
%O A168323 1,1
%A A168323 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A168322
%S A168322 3,5,5,7,7,17,13,23,17,31,19,47,23,61,29,67,31,83,37,103,41,107,43,113,
%T A168322 67,127,83,137,97,139,101,149,103,157,107,167,109,173,127,179,137,193,
%U A168322 149,199,151,227,163,229,179,233,181,239,193,241,197,263,199,271,239
%N A168322 a(1)=3,a(2)=5; a(n+1)=smallest prime number > a(n-1) such that the sum of any three consecutive terms is a prime.
%t A168322 a=3;b=5;lst={a,b};Do[Do[If[PrimeQ[q]&&PrimeQ[a+b+q],c=q;Break[]],{q,a+2,9!,2}];AppendTo[lst,c];a=b;b=c,{n,6!}];lst
%Y A168322 Cf. A062391
%K A168322 nonn,new
%O A168322 1,1
%A A168322 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A166998
%S A166998 1,0,6,28,2684,85664,96848424,18318978896,459531493100736,
%T A166998 468613553577122688,349607028167776160389536,
%U A166998 1788682277200384090414421312,46561932503015793339090359576558496
%N A166998 G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.
%F A166998 G.f.: sqrt([C(x)+S(x)]*[C(x)-S(x)]) where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.
%F A166998 Self-convolution yields A166998.
%e A166998 G.f: 1 + 6*x^2 + 28*x^3 + 2684*x^4 + 85664*x^5 + 96848424*x^6 +...
%e A166998 which equals sqrt( C(x)^2 - S(x)^2 ) where
%e A166998 C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
%e A166998 S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
%e A166998 Related expansions:
%e A166998 C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
%e A166998 C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
%o A166998 (PARI) {a(n)=polcoeff(sqrt(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2),n)}
%Y A166998 Cf. A166995, A166996, A166997, A060690, A014070.
%K A166998 nonn,new
%O A166998 0,3
%A A166998 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166997
%S A166997 1,0,12,56,5404,171664,193729840,36639136064,919064160383600,
%T A166997 937227332865348224,699214061851483321467008,
%U A166997 3577364560049979516493456896,93123865010226899737836259608990464
%N A166997 G.f.: C(x)^2 - S(x)^2 where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.
%F A166997 G.f.: [C(x)+S(x)]*[C(x)-S(x)] where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.
%F A166997 Self-convolution of A166998.
%e A166997 G.f: 1 + 12*x^2 + 56*x^3 + 5404*x^4 + 171664*x^5 + 193729840*x^6 +...
%e A166997 which equals C(x)^2 - S(x)^2 where
%e A166997 C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
%e A166997 S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
%e A166997 Related expansions:
%e A166997 C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
%e A166997 C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
%o A166997 (PARI) {a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2,n)}
%Y A166997 Cf. A166995, A166996, A166998, A060690, A014070.
%K A166997 nonn,new
%O A166997 0,3
%A A166997 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A168321
%S A168321 0,8,1,9,2,10,3,11,4,12,5,13,6,14,7,15,8,16,9,17,10,18,11,19,12,20,13,
%T A168321 21,14,22,15,23,16,24,17,25,18,26,19,27,20,28,21,29,22,30,23,31,24,32,
%U A168321 25,33,26,34,27,35,28,36,29,37,30,38,31,39,32,40,33,41,34,42,35,43,36
%N A168321 a(0)=7; a(n)=n-a(n-1).
%t A168321 a=7;Table[a=n-a,{n,a,200}]
%K A168321 nonn,new
%O A168321 1,2
%A A168321 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A166996
%S A166996 2,2,88,1028,289184,22451552,112890141568,50093449805856,
%T A166996 6676830881369059840,15354513520142235310592,
%U A166996 66620888067382334066280699904,750203718611121304644623635491840
%N A166996 G.f.: S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!, a power series in x with integer coefficients.
%e A166996 G.f: S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
%e A166996 The g.f. of A166995 is C(x):
%e A166996 C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!
%e A166996 C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
%e A166996 where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
%e A166996 and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
%e A166996 Related expansions:
%e A166996 C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
%e A166996 C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
%o A166996 (PARI) {a(n)=polcoeff(-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}
%Y A166996 Cf. A166995, A166997, A166998, A060690, A014070.
%K A166996 nonn,new
%O A166996 1,1
%A A166996 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166995
%S A166995 1,0,8,32,2848,87808,97425920,18364346368,459757145081856,
%T A166995 468713931103109120,349620381018764380930048,
%U A166995 1788712998645738038832398336,46562065744123901943395531497144320
%N A166995 G.f.: C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!, a power series in x with integer coefficients.
%e A166995 G.f: C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
%e A166995 The g.f. of A166996 is S(x):
%e A166995 S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!
%e A166995 S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
%e A166995 where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
%e A166995 and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
%e A166995 Related expansions:
%e A166995 C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
%e A166995 C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
%o A166995 (PARI) {a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!),n)}
%Y A166995 Cf. A166996, A166997, A166998, A060690, A014070.
%K A166995 nonn,new
%O A166995 0,3
%A A166995 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A168162
%S A168162 3,5,7,8,11,13,14,19,23,31,32,47,61
%N A168162 Numbers n which do not exceed the sum of the binary digits in all primes <= n.
%C A168162 The sequence A168161 is a subsequence of the primes in this sequence.
%F A168162 A168162 = { n | n <= A095375(pi(n)) }, where pi(n) = A000720(n).
%e A168162 There is no prime <= 1 and 2 has only nonzero binary digit, therefore these numbers are not in the sequence.
%e A168162 However, a(1)=3 has two binary digits, so the total number of these equal 3.
%e A168162 Then, 4 is larger than this, but the prime p=5 again adds 2 nonzero binary digits adding to a total of 5=a(2).
%e A168162 Then 6 is larger than this, but the prime p=7 adds 3 more nonzero bits for a total of 8, such that a(3)=7 and a(4)=8 don't exceed this.
%o A168162 (PARI) s=0; for(n=1,9999, isprime(n) && s+=norml2(binary(n)); n<=s & print1(n", "))
%K A168162 fini,full,nonn,new
%O A168162 1,1
%A A168162 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009

%I A168320
%S A168320 1,3,13,177,31345,982509057,965324047087029313,
%T A168320 931850515884481186273153523321252097,
%U A168320 868345383954173723655205051633111454677692184821907897236578371826897665
%N A168320 a(0)=0,a(n)=a^2+2^n.
%t A168320 a=0;Table[a=a^2+2^n,{n,0,11}]
%K A168320 nonn,new
%O A168320 1,2
%A A168320 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A168319
%S A168319 1,1,3,1,15,193,37185,1382724097,1911925928424465153,
%T A168319 3655460755781753047286544634111312897,
%U A168319 13362393337060505194526384126041275683735860286138280500673079957038531585
%V A168319 -1,-1,-3,1,-15,193,37185,1382724097,1911925928424465153,
%W A168319 3655460755781753047286544634111312897,
%X A168319 13362393337060505194526384126041275683735860286138280500673079957038531585
%N A168319 a(0)=0,a(n)=a^2-2^n.
%t A168319 a=0;Table[a=a^2-2^n,{n,0,11}]
%K A168319 nonn,new
%O A168319 1,3
%A A168319 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A168318
%S A168318 1,0,1,0,2,1,0,1,2,3,0,1,3,6,6,0,0,2,9,12,16,0,0,2,12,18,32,39,0,0,1,9,
%T A168318 24,48,78,103,0,0,1,9,30,64,117,206,263,0,0,0,6,24,80,156,309,526,690,0,
%U A168318 0,0,6,24,96,195,412,789,1380,1791
%N A168318 Triangle by rows, A168316 * its diagonalized eigensequence, A168317.
%C A168318 Row sums = A168317: (1, 1, 3, 6, 16, 39, 103, 263,...).
%C A168318 Rightmost diagonal = A168317 prefaced with a 1.
%C A168318 Sum of n-th row terms = rightmost term of next row.
%F A168318 Triangle by rows, M*Q. M = A168316, Q = an infinite lower triangular matrix
%F A168318 with A168317 prefaced with a 1; (1, 1, 1, 3, 6, 16, 39, 103,...) as the right
%F A168318 diagonal and the rest zeros.
%e A168318 First few rows of the triangle =
%e A168318 1;
%e A168318 0, 1;
%e A168318 0, 2, 1;
%e A168318 0, 1, 2, 3;
%e A168318 0, 1, 3, 6, 6;
%e A168318 0, 0, 2, 9, 12, 16;
%e A168318 0, 0, 2, 12, 18, 32, 39;
%e A168318 0, 0, 1, 9, 24, 48, 78, 103;
%e A168318 0, 0, 1, 9, 30, 64, 117, 206, 263;
%e A168318 0, 0, 0, 6, 24, 80, 156, 309, 526, 690;
%e A168318 0, 0, 0, 6, 24, 96, 195, 412, 789, 1380, 1791;
%e A168318 0, 0, 0, 3, 18, 80, 234, 515, 1052, 2070, 3582, 4693;
%e A168318 0, 0, 0, 3, 18, 80, 273, 618, 1315, 2760, 5373, 9386, 12247;
%e A168318 0, 0, 0, 0, 12, 64, 234, 721, 1578, 3450, 7164, 14079, 24494, 32073;
%e A168318 ...
%Y A168318 Cf. A168316, A168317
%K A168318 nonn,tabl,new
%O A168318 1,5
%A A168318 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168317
%S A168317 1,1,3,6,16,39,103,263,690,1791,4693,12247,32073,83869,219598,574658,
%T A168317 1504540,3938272,10310703
%N A168317 Eigensequence of triangle A168316
%C A168317 Conjectured convergent of a(n)/a(n-1) = phi^2, 2.6180339...
%C A168317 a(19)/a(18) = 10310703/3938272 = 2.6180779....
%F A168317 Let M = triangle A168316 as an infinite lower triangular matrix. Shift M
%F A168317 down one row, inserting a "1" at top. A168317 = Lim_{n->inf.} M^2, the
%F A168317 left shifted vector considered as a sequence.
%Y A168317 Cf. A168316, A168318
%K A168317 eigen,nonn,new
%O A168317 1,3
%A A168317 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168316
%S A168316 1,0,1,0,2,1,0,1,2,1,0,1,3,2,1,0,0,2,3,2,1,0,0,2,4,3,2,1,0,0,1,3,4,3,2,
%T A168316 1,0,0,1,3,5,4,3,2,1,0,0,0,2,4,5,4,3,2,1,0,0,2,4,6,5,4,3,2,1,0,0,1,3,5,
%U A168316 6,5,4,3,2,1,0,0,0,1,3,5,7,6,5,4,3,2,1
%N A168316 Triangle by rows, square of triangle A101688
%C A168316 Row sums = A129819 starting (1, 1, 3, 4, 7, 8, 12,...).
%C A168316 Eigensequence of the triangle = A168317: (1, 1, 3, 6, 16, 39, 103, 263, 690,...).
%F A168316 Triangle by rows, (A101688)^2, as an infinite lower triangular matrix.
%e A168316 First few rows of the triangle =
%e A168316 1;
%e A168316 0, 1;
%e A168316 0, 2, 1;
%e A168316 0, 1, 2, 1;
%e A168316 0, 1, 3, 2, 1;
%e A168316 0, 0, 2, 3, 2, 1;
%e A168316 0, 0, 2, 4, 3, 2, 1;
%e A168316 0, 0, 1, 3, 4, 3, 2, 1;
%e A168316 0, 0, 0, 2, 4, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 2, 4, 6, 5, 4, 3, 2, 1
%e A168316 0, 0, 0, 1, 3, 5, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 1, 3, 5, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 2, 4, 6, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 2, 4, 6, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 1, 3, 5, 7, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 1, 3, 5, 7, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 0, 2, 4, 6, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 0, 0, 0, 0, 0, 1, 3, 5, 7, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168316 ...
%Y A168316 Cf. A101688, A129819, A168317, A168318
%K A168316 nonn,tabl,new
%O A168316 1,5
%A A168316 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168315
%S A168315 1,0,1,0,2,1,0,0,2,3,0,0,2,6,5,0,0,0,6,10,13,0,0,0,6,10,26,29,0,0,0,0,
%T A168315 10,26,58,71,0,0,0,0,10,26,58,142,165,0,0,0,0,0,26,58,142,330,401,0,0,0,
%U A168315 0,0,26,58,142,330,802,957
%N A168315 Triangle by rows, A168313 * the diagonalized variant of its eigensequence, A168314.
%C A168315 Row sums = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401, 957,...).
%C A168315 Rightmost column = A168314 prefaced with a 1.
%C A168315 Sum of n-th row terms = rightmost term of next row.
%F A168315 Let M = triangle A168313 and Q = in an infinite lower triangular matrix with
%F A168315 A168314 prefaced with a 1 as the rightmost diagonal with the rest of terms 0's.
%F A168315 Triangle A168315 = M*Q.
%e A168315 First few rows of the triangle =
%e A168315 1;
%e A168315 0, 1;
%e A168315 0, 2, 1;
%e A168315 0, 0, 2, 3;
%e A168315 0, 0, 2, 6, 5;
%e A168315 0, 0, 0, 6, 10, 13;
%e A168315 0, 0, 0, 6, 10, 26, 29;
%e A168315 0, 0, 0, 6, 10, 26, 58, 71;
%e A168315 0, 0, 0, 0, 10, 26, 58, 142, 165;
%e A168315 0, 0, 0, 0, 0, 26, 58, 142, 330, 401;
%e A168315 0, 0, 0, 0, 0, 26, 58, 142, 330, 802, 957;
%e A168315 0, 0, 0, 0, 0, 0, 58, 142, 330, 802, 1914, 2315;
%e A168315 0, 0, 0, 0, 0, 0, 58, 142, 330, 802, 1914, 4630, 5561;
%e A168315 ...
%Y A168315 Cf. A168313, A168314
%K A168315 nonn,tabl,new
%O A168315 1,5
%A A168315 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168314
%S A168314 1,1,3,5,13,29,71,165,401,957,2315,5561,13437,32377,78191,188617,455425,
%T A168314 1099137,2653699
%N A168314 Eigensequence of triangle A168313
%C A168314 Conjectured convergent of a(n)/a(n-1) = (1+sqrt(2)). a(19)a(18) =
%C A168314 2653699/1099137 = 2.4143478...
%F A168314 Let M = A168313 shifted down one row and inserting a "1" at top.
%F A168314 A168314 = the eigensequence of triangle A168313 = Lim_{n->inf.} M^n;
%F A168314 = the left column vector as a sequence.
%Y A168314 Cf. A168313, A168315
%K A168314 eigen,nonn,new
%O A168314 1,3
%A A168314 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168313
%S A168313 1,0,1,0,2,1,0,0,2,1,0,0,2,2,1,0,0,0,2,2,1,0,0,0,2,2,2,1,0,0,0,0,2,2,2,
%T A168313 1,0,0,0,0,2,2,2,2,1,0,0,0,0,0,2,2,2,2,1,0,0,0,0,0,2,2,2,2,2,1
%N A168313 Triangle by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
%C A168313 Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).
%C A168313 Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).
%F A168313 Triangle by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
%e A168313 First few rows of the triangle =
%e A168313 1;
%e A168313 0, 1;
%e A168313 0, 2, 1;
%e A168313 0, 0, 2, 1;
%e A168313 0, 0, 2, 2, 1;
%e A168313 0, 0, 0, 2, 2, 1;
%e A168313 0, 0, 0, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
%e A168313 ...
%Y A168313 Cf. A101688, A168314, A168315
%K A168313 nonn,tabl,new
%O A168313 1,5
%A A168313 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168312
%S A168312 1,2,1,0,0,1,4,2,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,1,8,4,0,2,0,0,0,
%T A168312 1,0,0,0,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,
%U A168312 2,0,0,0,0,0,1
%N A168312 Triangle by rows, replace 1's of triangle A115361 starting from the right with (1, 2, 4, 8,...).
%C A168312 Row sums = A038712: (1, 3, 1, 7, 1, 3, 1, 15,...).
%F A168312 Triangle by rows, replace 1's in triangle A115361 starting from the right with
%F A168312 (1, 2, 4, 8,...).
%e A168312 First few rows of the triangle =
%e A168312 1;
%e A168312 2, 1;
%e A168312 0, 0, 1;
%e A168312 4, 2, 0, 1;
%e A168312 0, 0, 0, 0, 1;
%e A168312 0, 0, 2, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 1;
%e A168312 8, 4, 0, 2, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 2, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 16, 8, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A168312 ...
%Y A168312 Cf. A115361, A038712
%K A168312 nonn,tabf,new
%O A168312 1,2
%A A168312 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009

%I A168311
%S A168311 33,41,53,54,63,75,86,96,104,117,118,125,129,162,167,179,180,185,188,
%T A168311 195,204,222,223,224,229,230,243,248,251,261,270,271,284,285,293,294,
%U A168311 314,317,318,333,334,338,339,348,349,350,360,365,369,375,376,377,383
%N A168311 Numbers that are not sums of two triangular numbers and are not one greater than a sum of two triangular numbers.
%C A168311 Number n belongs to this sequence if both n and n-1 belong to A020757.
%Y A168311 A020757
%K A168311 nonn,new
%O A168311 1,1
%A A168311 Tanya Khovanova (tanyakh(AT)yahoo.com), Nov 22 2009

%I A168310
%S A168310 8,800,808,818,880,888,885,884,889,881
%N A168310 The first thousand positive integers arranged alphabetically
%C A168310 This is a poem by Claude Closky.
%H A168310 <a href="http://www.ubu.com/concept/closky_1000.html"> Poem on Ubuweb </a>
%K A168310 nonn,new
%O A168310 1,1
%A A168310 Douglas Summers-Stay (dss316(AT)nyu.edu), Nov 22 2009

%I A168309
%S A168309 4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,
%T A168309 4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,
%U A168309 4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3
%V A168309 4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,
%W A168309 4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,
%X A168309 4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3,4,-3
%N A168309 Period 2: repeat 4,-3.
%C A168309 Interleaving of A010709 and -3*A000012.
%C A168309 Binomial transform of 4 followed by a signed version of A005009.
%C A168309 Inverse binomial transform of 4 followed by A000079.
%C A168309 a(n+1)-a(n) = 7*(-1)^n.
%C A168309 A168230 without initial term 0 gives partial sums.
%F A168309 a(n) = (-7*(-1)^n+1)/2.
%F A168309 a(n) = -a(n-1)+1 for n > 1; a(1) = 4.
%F A168309 a(n) = a(n-2) for n > 2; a(1) = 4, a(2) = -3.
%F A168309 G.f.: x*(4-3*x)/((1-x)*(1+x)).
%o A168309 (MAGMA) &cat[ [4, -3]: n in [1..42] ];
%o A168309 [ n eq 1 select 4 else -Self(n-1)+1: n in [1..84] ];
%Y A168309 Cf. A010709 (all 4's sequence), A000012 (all 1's sequence), A010727 (all 7's sequence), A168230, A005009 (7*2^n), A000079 (powers of 2).
%K A168309 sign,new
%O A168309 1,1
%A A168309 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 22 2009

%I A168301
%S A168301 1,7,7,13,13,19,19,25,25,31,31,37,37,43,43,49,49,55,55,61,61,67,67,73,
%T A168301 73,79,79,85,85,91,91,97,97,103,103,109,109,115,115,121,121,127,127,133,
%U A168301 133,139,139,145,145,151,151,157,157,163,163,169,169,175,175,181,181
%N A168301 a(n)=6*n-a(n-1)-4 (with a(1)=1)
%F A168301 a(n)=6*n-a(n-1)-4 (with a(1)=1)
%e A168301 For n=2, a(2)=6*2-1-4=7; n=3, a(3)=6*3-7-4=7; n=4, a(4)=6*4-7-4=13
%K A168301 nonn,new
%O A168301 1,2
%A A168301 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168300
%S A168300 5,5,11,11,17,17,23,23,29,29,35,35,41,41,47,47,53,53,59,59,65,65,71,71,
%T A168300 77,77,83,83,89,89,95,95,101,101,107,107,113,113,119,119,125,125,131,
%U A168300 131,137,137,143,143,149,149,155,155,161,161,167,167,173,173,179,179
%N A168300 a(n)=6*n-a(n-1)-2 (with a(1)=5)
%F A168300 a(n)=6*n-a(n-1)-2 (with a(1)=5)
%e A168300 For n02, a(2)=6*2-5-2=5; n=3, a(3)=6*3-5-2=11; n=4, a(4)=6*4-11-2=11
%K A168300 nonn,new
%O A168300 1,1
%A A168300 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168299
%S A168299 1,4,73,730,5185,30376,157465,750142,3359233,14348908,59049001,
%T A168299 235782658,918330049,3502727632,13124466937,48427561126,176319369217,
%U A168299 634465620820,2259436291849,7971951402154,27894275208001,96873331012984
%N A168299 Numerator(3^n-n^3) {n=0...-infinity}.
%t A168299 f[n_]:=3^n-n^3;Table[Numerator[f[n]],{n,0,-50,-1}]
%Y A168299 Cf. A024026
%K A168299 nonn,new
%O A168299 1,2
%A A168299 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A168298
%S A168298 1,1,15,71,255,799,2303,6271,16383,41471,102399,247807,
%T A168298 589823,1384447,3211263,7372799,16777215,37879807,84934655,
%U A168298 189267967,419430399,924844031,2030043135,4437573631,9663676415
%V A168298 1,-1,-15,-71,-255,-799,-2303,-6271,-16383,-41471,-102399,-247807,
%W A168298 -589823,-1384447,-3211263,-7372799,-16777215,-37879807,-84934655,
%X A168298 -189267967,-419430399,-924844031,-2030043135,-4437573631,-9663676415
%N A168298 2^n-n^2 {n=0...-infinity}.
%t A168298 f[n_]:=2^n-n^2;Table[Numerator[f[n]],{n,0,-50,-1}]
%Y A168298 Cf. A024012
%K A168298 nonn,new
%O A168298 1,3
%A A168298 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A168297
%S A168297 1,1,9,31,73,141,241,379,561,793,1081,1431,1849,2341,
%T A168297 2913,3571,4321,5169,6121,7183,8361,9661,11089,12651,14353,
%U A168297 16201,18201,20359,22681,25173,27841,30691,33729,36961,40393
%V A168297 -1,-1,-9,-31,-73,-141,-241,-379,-561,-793,-1081,-1431,-1849,-2341,
%W A168297 -2913,-3571,-4321,-5169,-6121,-7183,-8361,-9661,-11089,-12651,-14353,
%X A168297 -16201,-18201,-20359,-22681,-25173,-27841,-30691,-33729,-36961,-40393
%N A168297 a(n)=n^3-(n+1)^2 {n=0...-infinity}.
%t A168297 f[n_]:=n^3-(n+1)^2;Table[f[n],{n,0,-50,-1}]
%Y A168297 Cf. A153257
%K A168297 nonn,new
%O A168297 1,3
%A A168297 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 22 2009

%I A166893
%S A166893 1,6,63,986,20685,545736,17365336,647216568,27653205177,1332422277828,
%T A166893 71470510481961,4223498675806638,272615162534575302,
%U A166893 19082609490868539738,1439738711122827542742,116468234559061615308870
%N A166893 Column 3 of triangle A166890.
%C A166893 Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.
%o A166893 (PARI) {a(n)=local(F=x, M, N, P, m=n+2); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+2, 3]}
%Y A166893 Cf. A166890, A166891, A166892.
%K A166893 nonn,new
%O A166893 1,2
%A A166893 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166892
%S A166892 1,4,30,364,6233,139008,3833052,126105168,4824243516,210489178476,
%T A166892 10318212622770,561491367744672,33588989930164050,2190978413703916624,
%U A166892 154771816676784778818,11771103512077651149912,959000166676677798631894
%N A166892 Column 2 of triangle A166890.
%C A166892 Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.
%o A166892 (PARI) {a(n)=local(F=x, M, N, P, m=n+1); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, 2]}
%Y A166892 Cf. A166890, A166891, A166893.
%K A166892 nonn,new
%O A166892 1,2
%A A166892 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166891
%S A166891 1,2,9,78,1038,18968,443595,12681960,429244197,16801151910,746998729887,
%T A166891 37200237947376,2051666003699226,124156748403386646,8180285024067867345,
%U A166891 582970677419310149580,44684461723038752605932
%N A166891 Column 1 of triangle A166890.
%C A166891 Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.
%o A166891 (PARI) {a(n)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n, 1]}
%Y A166891 Cf. A166890, A166892, A166893.
%K A166891 nonn,new
%O A166891 1,2
%A A166891 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A168111
%S A168111 0,1,1,3,1,6,1,8,4,10,1,22
%N A168111 A047968(n) - A000041(n).
%C A168111 Row sums of triangle A168021 except the first column.
%C A168111 Row sums of triangle A168016 except the last column.
%Y A168111 Cf. A000041, A047968, A168016, A168017, A168018, A168020, A168021.
%K A168111 easy,more,nonn,new
%O A168111 1,4
%A A168111 Omar E. Pol (info(AT)polprimos.com), Nov 22 2009

%I A166890
%S A166890 1,2,1,9,4,1,78,30,6,1,1038,364,63,8,1,18968,6233,986,108,10,1,443595,
%T A166890 139008,20685,2072,165,12,1,12681960,3833052,545736,51494,3750,234,14,1,
%U A166890 429244197,126105168,17365336,1569920,107760,6148,315,16,1,16801151910
%N A166890 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x*(1+x)^2 (cf. A166888).
%e A166890 Triangle begins:
%e A166890 1;
%e A166890 2,1;
%e A166890 9,4,1;
%e A166890 78,30,6,1;
%e A166890 1038,364,63,8,1;
%e A166890 18968,6233,986,108,10,1;
%e A166890 443595,139008,20685,2072,165,12,1;
%e A166890 12681960,3833052,545736,51494,3750,234,14,1;
%e A166890 429244197,126105168,17365336,1569920,107760,6148,315,16,1;
%e A166890 16801151910,4824243516,647216568,56661004,3728952,200583,9394,408,18,1;
%e A166890 746998729887,210489178476,27653205177,2361036896,150566205,7768320,343063,13616,513,20,1;
%e A166890 37200237947376,10318212622770,1332422277828,111501524409,6938694600,347030328,14703080,550300,18942,630,22,1; ...
%e A166890 Coefficients in iterations of x*(1+x)^2 form table A166888:
%e A166890 1;
%e A166890 1,2,1;
%e A166890 1,4,10,18,23,22,15,6,1;
%e A166890 1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
%e A166890 1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
%e A166890 1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
%e A166890 1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
%e A166890 1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
%e A166890 1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
%e A166890 ...
%e A166890 This triangle T transforms one diagonal in A166888 into another,
%e A166890 for example: T * A154256 = A119820, T * A119820 = A166889, where
%e A166890 A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
%e A166890 A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
%e A166890 A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
%o A166890 (PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%Y A166890 Cf. columns: A166891, A166892, A166893, variants: A135080, A166884.
%Y A166890 Cf. A166888, A154256, A119820, A166889.
%K A166890 nonn,tabl,new
%O A166890 1,2
%A A166890 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166889
%S A166889 1,6,52,660,11385,251832,6842220,221228244,8311401351,356190316416,
%T A166889 17160064580802,918453056609946,54085054802995008,3475794779752572784,
%U A166889 242103490865991893116,18170143514998451547348
%N A166889 Coefficients of x^n in the (n+1)-th iteration of x*(1+x)^2 for n>=1.
%e A166889 Coefficients in the initial iterations of x*(1+x)^2 begin:
%e A166889 [1,2,1];
%e A166889 [(1),4,10,18,23,22,15,6,1];
%e A166889 [1,(6),27,102,333,960,2472,5748,12150,23388,...];
%e A166889 [1,8,(52),300,1578,7692,35094,150978,615939,2393628,...];
%e A166889 [1,10,85,(660),4790,32920,215988,1360638,8265613,48585702,...];
%e A166889 [1,12,126,1230,(11385),101010,864813,7178700,57976074,...];
%e A166889 [1,14,175,2058,23163,(251832),2660028,27405798,276215313,...];
%e A166889 [1,16,232,3192,42308,544600,(6842220),84191772,1017153322,...];
%e A166889 [1,18,297,4680,71388,1061712,15463512,(221228244),3115739358,...];
%e A166889 [1,20,370,6570,113355,1912590,31683051,516686346,(8311401351),...]; ...
%e A166889 where the coefficients in parenthesis form the initial terms of this sequence.
%o A166889 (PARI) {a(n)=local(F=x*(1+x)^2, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n, x)))}
%Y A166889 Cf. A166888, A154256, A119820, A166890.
%K A166889 nonn,new
%O A166889 1,2
%A A166889 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A166888
%S A166888 1,1,2,1,1,4,10,18,23,22,15,6,1,1,6,27,102,333,960,2472,5748,12150,
%T A166888 23388,40926,64872,92772,119216,137112,140526,127677,102150,71331,42954,
%U A166888 21939,9288,3156,822,153,18,1,1,8,52,300,1578,7692,35094,150978
%N A166888 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of x*(1+x)^2.
%e A166888 Triangle begins:
%e A166888 1;
%e A166888 1,2,1;
%e A166888 1,4,10,18,23,22,15,6,1;
%e A166888 1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
%e A166888 1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
%e A166888 1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
%e A166888 1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
%e A166888 1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
%e A166888 1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
%e A166888 1,18,297,4680,71388,1061712,15463512,221228244,3115739358,...;
%e A166888 1,20,370,6570,113355,1912590,31683051,516686346,8311401351,...;
%e A166888 1,22,451,8910,171545,3237520,60108576,1100544720,19906483168,...;
%e A166888 1,24,540,11748,249678,5211492,107184066,2176952910,43733857365,...;
%e A166888 ...
%e A166888 The initial diagonals in this triangle begin:
%e A166888 A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
%e A166888 A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
%e A166888 A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
%e A166888 The diagonals are transformed one into the other by
%e A166888 triangle A166890, which begins:
%e A166888 1;
%e A166888 2,1;
%e A166888 9,4,1;
%e A166888 78,30,6,1;
%e A166888 1038,364,63,8,1;
%e A166888 18968,6233,986,108,10,1;
%e A166888 443595,139008,20685,2072,165,12,1;
%e A166888 12681960,3833052,545736,51494,3750,234,14,1; ...
%o A166888 (PARI) {T(n, k)=local(F=x+2*x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
%Y A166888 Cf. diagonals: A154256, A119820, A166889, variants: A166880, A122888.
%K A166888 nonn,tabf,new
%O A166888 0,3
%A A166888 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2009

%I A168296
%S A168296 1,1,2,2,18,18,6,156,432,288,24,792,7416,13248,6624,120,11280,64800,
%T A168296 374400,496800,198720,720,62640,1254960,4968000,20865600,22057920,
%U A168296 7352640,5040,24012000,11854080,125677440,389491200,1288103040
%V A168296 1,1,2,2,18,18,6,156,432,288,24,792,7416,13248,6624,120,-11280,64800,
%W A168296 374400,496800,198720,720,-62640,-1254960,4968000,20865600,22057920,
%X A168296 7352640,5040,24012000,-11854080,-125677440,389491200,1288103040
%N A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
%C A168296 Row sums are:
%C A168296 {1, 3, 38, 882, 28104, 1123560, 53927280, 3019902480, 193273557120, 13915694298240,...}.
%C A168296 In Comtet there is this function:
%C A168296 x^n=Sum[Eulerian[n,k*Binomial[x+k-1,n],{k,1,n]]
%C A168296 In OEIS I was looking for an Umbral Calculus expansion for the MacMahon and
%C A168296 found this "Worpitzky form":
%C A168296 Sum [MacMahon[n,k]*Binomial[x+k-1,n-1],{k,1,n}]=(2*x+1)^(n+1)
%C A168296 The use the infinite sums k, 2*k+1 type polynomials
%C A168296 and are pretty much alike except for a sliding offset in n.
%C A168296 Conjecture: "Worpitzky forms"
%C A168296 Some general polynomial form:general Pascal recursion Pascal[n,k,m]
%C A168296 p[x,n,m]=Sum [Pascal[n,k,m]*Binomial[x+k-1,n-1],{k,1,n}]
%C A168296 where p[x,n,m] are the inverse z transform polynomials.
%F A168296 p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
%e A168296 {1},
%e A168296 {1, 2},
%e A168296 {2, 18, 18},
%e A168296 {6, 156, 432, 288},
%e A168296 {24, 792, 7416, 13248, 6624},
%e A168296 {120, -11280, 64800, 374400, 496800, 198720},
%e A168296 {720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640},
%e A168296 {5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160},
%e A168296 {40320, 192378240, 5004581760, -1669248000, -12569437440, 32116331520, 87702289920, 65997296640, 16499324160},
%e A168296 {362880, -119545632000, 57161064960, 868954106880, -218287560960, -1293900894720, 2812649495040, 6545378949120, 4306323605760, 956960801280}
%t A168296 (*Worpitzky form polynomials for A142462*)
%t A168296 Clear[A, m, n, k, a, p]
%t A168296 m = 7;
%t A168296 A[n_, 1] := 1 A[n_, n_] := 1
%t A168296 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k];
%t A168296 a = Table[A[n, k], {n, 10}, {k, n}];
%t A168296 p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];
%t A168296 Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];
%t A168296 Flatten[%]
%Y A168296 A142462
%K A168296 nonn,uned,new
%O A168296 1,3
%A A168296 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168295
%S A168295 1,1,2,2,10,10,6,52,120,80,24,280,1160,1760,880,120,1520,10000,27200,
%T A168295 30800,12320,720,11280,78160,343200,695200,628320,209440,5040,164640,
%U A168295 784000,3684800,12073600,19490240,14660800,4188800,40320,1438080
%N A168295 Worpitzky form polynomials for the {1,8,1} A142458 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
%C A168295 Row sums are:
%C A168295 {1, 3, 22, 258, 4104, 81960, 1966320, 55051920, 1761621120, 63417997440,...}.
%C A168295 Dividing row A167786 by 3^n gets a very similar sequence.
%C A168295 In Comtet there is this function:
%C A168295 x^n=Sum[Eulerian[n,k*Binomial[x+k-1,n],{k,1,n]]
%C A168295 In OEIS I was looking for an Umbral Calculus expansion for the MacMahon and
%C A168295 found this "Worpitzky form":
%C A168295 Sum [MacMahon[n,k]*Binomial[x+k-1,n-1],{k,1,n}]=(2*x+1)^(n+1)
%C A168295 The use the infinite sums k, 2*k+1 type polynomials
%C A168295 and are pretty much alike except for a sliding offset in n.
%C A168295 Conjecture: "Worpitzky forms"
%C A168295 Some general polynomial form:general Pascal recursion Pascal[n,k,m]
%C A168295 p[x,n,m]=Sum [Pascal[n,k,m]*Binomial[x+k-1,n-1],{k,1,n}]
%C A168295 where p[x,n,m] are the inverse z transform polynomials.
%F A168295 p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
%e A168295 {1},
%e A168295 {1, 2},
%e A168295 {2, 10, 10},
%e A168295 {6, 52, 120, 80},
%e A168295 {24, 280, 1160, 1760, 880},
%e A168295 {120, 1520, 10000, 27200, 30800, 12320},
%e A168295 {720, 11280, 78160, 343200, 695200, 628320, 209440},
%e A168295 {5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800},
%e A168295 {40320, 1438080, 15532160, 48294400, 170755200, 445688320, 598160640, 385369600, 96342400},
%e A168295 {362880, -51206400, 178617600, 1217036800, 2840745600, 8032738560, 17417030400, 20005708800, 11272060800, 2504902400}
%t A168295 (*Worpitzky form polynomials for A142458*)
%t A168295 Clear[A, m, n, k, a, p]
%t A168295 m = 3;
%t A168295 A[n_, 1] := 1 A[n_, n_] := 1
%t A168295 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k];
%t A168295 a = Table[A[n, k], {n, 10}, {k, n}];
%t A168295 p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];
%t A168295 Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];
%t A168295 Flatten[%]
%Y A168295 A167786, A142458
%K A168295 nonn,uned,new
%O A168295 1,3
%A A168295 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168294
%S A168294 0,1,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A168294 131072,262144,524288,1048576
%N A168294 Terms of rank 0,1,2,4,8,16,=A131577 of A022998=0,1,4,3,8,5,12,7,16,.
%p A168294 a(n)=0,A151821 = A131577 without 2.
%K A168294 nonn,uned,new
%O A168294 0,3
%A A168294 Paul Curtz (bpcrtz(AT)free.fr), Nov 22 2009

%I A168293
%S A168293 1,1,1,1,14,1,1,33,33,1,1,64,186,64,1,1,119,724,724,119,1,1,222,2415,
%T A168293 5120,2415,222,1,1,421,7491,28799,28799,7491,421,1,1,812,22456,142268,
%U A168293 257866,142268,22456,812,1,1,1587,66342,649554,1934544,1934544,649554
%N A168293 Coefficients of the expansion of:w=3/2;p(t,x)=-4*((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) + (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))
%C A168293 A umbral calculus expansion made from a double quadratic Bezier extrapolation of the Pascal,Eulerian and A046802.
%C A168293 The first term is adjusted to 1.
%C A168293 Row sums are:
%C A168293 {1, 2, 16, 68, 316, 1688, 10396, 73424, 588940, 5304056, 530487642,...}
%F A168293 w=3/2;p(t,x)=-4*((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) +
%F A168293 (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))
%e A168293 {1},
%e A168293 {1, 1},
%e A168293 {1, 14, 1},
%e A168293 {1, 33, 33, 1},
%e A168293 {1, 64, 186, 64, 1},
%e A168293 {1, 119, 724, 724, 119, 1},
%e A168293 {1, 222, 2415, 5120, 2415, 222, 1},
%e A168293 {1, 421, 7491, 28799, 28799, 7491, 421, 1},
%e A168293 {1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1},
%e A168293 {1, 1587, 66342, 649554, 1934544, 1934544, 649554, 66342, 1587, 1},
%e A168293 {1, 3130, 195093, 2822556, 12915534, 21176136, 12915534, 2822556, 195093, 3130, 1}
%t A168293 w=3/2;
%t A168293 p[t_] =-4*((1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[t])/(1-x* Exp[t*(1-x)])+w^2*(1-x)/(x*(1- x*Exp[t*(1-x)]))+
%t A168293 (1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[ t])/(1-x*Exp[t*(1-x)])+w^2*(1-x)/(1-x*Exp[t*(1-x)]))
%t A168293 a = Join[ {{1}}, Table[ CoefficientList[ FullSimplify[ ExpandAll[ n!*SeriesCoefficient[ Series[ \ p[ t ], {t, 0, 30} ], n ] ] ], x ], {n, 1, 10} ] ];
%t A168293 Flatten[a]
%Y A168293 A046802
%K A168293 nonn,uned,new
%O A168293 0,5
%A A168293 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168292
%S A168292 7,7,7,7,38,7,7,99,99,7,7,220,546,220,7,7,461,2236,2236,461,7,7,942,
%T A168292 8001,15596,8001,942,7,7,1903,26697,89921,89921,26697,1903,7,7,3824,
%U A168292 85660,463520,796594,463520,85660,3824,7,7,7665,268530,2224350,6068400
%N A168292 Coefficients of the expansion of:w=3;p(t,x)=-((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) + (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))
%C A168292 A umbral calculus expansion made from a double quadratic Bezier extrapolation of the Pascal,Eulerian and A046802.
%C A168292 The first term is adjusted to 7.
%C A168292 Row sums are:
%C A168292 {7, 14, 52, 212, 1000, 5408, 33496, 237056, 1902616, 17137904, 171411832,...}
%F A168292 w=3;p(t,x)=-((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) +
%F A168292 (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))
%e A168292 {7},
%e A168292 {7, 7},
%e A168292 {7, 38, 7},
%e A168292 {7, 99, 99, 7},
%e A168292 {7, 220, 546, 220, 7},
%e A168292 {7, 461, 2236, 2236, 461, 7},
%e A168292 {7, 942, 8001, 15596, 8001, 942, 7},
%e A168292 {7, 1903, 26697, 89921, 89921, 26697, 1903, 7},
%e A168292 {7, 3824, 85660, 463520, 796594, 463520, 85660, 3824, 7},
%e A168292 {7, 7665, 268530, 2224350, 6068400, 6068400, 2224350, 268530, 7665, 7},
%e A168292 {7, 15346, 829683, 10171920, 41720142, 65937636, 41720142, 10171920, 829683, 15346, 7}
%t A168292 w=3;
%t A168292 p[t_] =-((1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[t])/(1-x* Exp[t*(1-x)])+w^2*(1-x)/(x*(1- x*Exp[t*(1-x)]))+
%t A168292 (1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[ t])/(1-x*Exp[t*(1-x)])+w^2*(1-x)/(1-x*Exp[t*(1-x)]))
%t A168292 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168292 Flatten[a]
%Y A168292 A046802
%K A168292 nonn,uned,new
%O A168292 0,1
%A A168292 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168291
%S A168291 1,1,1,1,6,1,1,15,15,1,1,32,82,32,1,1,65,330,330,65,1,1,130,1159,2304,
%T A168291 1159,130,1,1,259,3801,13195,13195,3801,259,1,1,516,12016,67316,117170,
%U A168291 67316,12016,516,1,1,1029,37212,319332,889230,889230,319332,37212,1029
%N A168291 Coefficients of the expansion of:w=2;p(t,x)=-((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) + (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))/2
%C A168291 A umbral calculus expansion made from a double quadratic Bezier extrapolation of the Pascal,Eulerian and A046802.
%C A168291 The first term is adjusted to 1.
%C A168291 Row sums are:
%C A168291 {1, 2, 8, 32, 148, 792, 4884, 34512, 276868, 2493608, 24940180,...}
%F A168291 w=2;p(t,x)=-((1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)(1 - x)*(Exp[ t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/( x*(1 - x*Exp[t*(1 - x)])) +
%F A168291 (1 - w)^2Exp[t*(1 + x)] + 2*w*(1 - w)( 1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)]) + w^2*(1 - x)/(1 - x*Exp[t*(1 - x)]))/2
%e A168291 {1},
%e A168291 {1, 1},
%e A168291 {1, 6, 1},
%e A168291 {1, 15, 15, 1},
%e A168291 {1, 32, 82, 32, 1},
%e A168291 {1, 65, 330, 330, 65, 1},
%e A168291 {1, 130, 1159, 2304, 1159, 130, 1},
%e A168291 {1, 259, 3801, 13195, 13195, 3801, 259, 1},
%e A168291 {1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1},
%e A168291 {1, 1029, 37212, 319332, 889230, 889230, 319332, 37212, 1029, 1},
%e A168291 {1, 2054, 113869, 1443844, 6070654, 9679336, 6070654, 1443844, 113869, 2054, 1}
%t A168291 w=2;
%t A168291 p[t_] =-((1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[t])/(1-x* Exp[t*(1-x)])+w^2*(1-x)/(x*(1- x*Exp[t*(1-x)]))+
%t A168291 (1-w)^2Exp[t*(1+x)]+2*w*(1-w)(1-x)*(Exp[ t])/(1-x*Exp[t*(1-x)])+w^2*(1-x)/(1-x*Exp[t*(1-x)]))/2
%t A168291 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168291 Flatten[a]
%Y A168291 A046802
%K A168291 nonn,uned,new
%O A168291 0,5
%A A168291 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168017
%S A168017 1,1,2,1,3,1,2,5,1,7,1,2,3,11,1,15,1,2,5,22,1,3,30,1,2,7,42,1,56,1,2,3,
%T A168017 5,11,77
%N A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.
%C A168017 Positive values of triangle A168016.
%C A168017 The number of terms of row n is equal to the number of divisors of n: A000005(n).
%C A168017 Note that the last term of each row is the number of partitions of n: A000041(n).
%H A168017 O. E. Pol,<a href="www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the partitions of n, for n = 1 .. 9</a>
%e A168017 For example:
%e A168017 Consider the row 8: (1, 2, 5, 22). The divisor of 8 listed in decreasing order are 8, 4, 2, 1 (See A056538). Also, there is 1 partitions of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}. Also, there are 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}. Finally, there are 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then the row 8 is formed by 1, 2, 5, 22.
%e A168017 Triangle begins:
%e A168017 1;
%e A168017 1, 2;
%e A168017 1, 3;
%e A168017 1, 2, 5;
%e A168017 1, 7;
%e A168017 1, 2, 3, 11;
%e A168017 1, 15;
%e A168017 1, 2, 5, 22;
%e A168017 1, 3, 30;
%e A168017 1, 2, 7, 42;
%e A168017 1, 56;
%e A168017 1, 2, 3, 5, 11, 77;
%Y A168017 Row sums give A047968.
%Y A168017 Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.
%K A168017 more,nonn,tabf,new
%O A168017 1,3
%A A168017 Omar E. Pol (info(AT)polprimos.com), Nov 22 2009

%I A168290
%S A168290 1,1,1,1,7,1,1,23,23,1,1,59,141,59,1,1,135,615,615,135,1,1,291,2305,
%T A168290 4335,2305,291,1,1,607,7971,25415,25415,7971,607,1,1,1243,26293,133771,
%U A168290 224365,133771,26293,1243,1,1,2519,84191,656039,1722251,1722251,656039
%N A168290 Coefficients of the expansion of:w=5;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%C A168290 A umbral calculus expansion made from a Bezier extrapolation of the Pascal and A046802.
%C A168290 Row sums are:
%C A168290 {1, 2, 9, 48, 261, 1502, 9529, 67988, 546981, 4930002, 49316409,...}
%F A168290 w=5;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%e A168290 {1},
%e A168290 {1, 1},
%e A168290 {1, 7, 1},
%e A168290 {1, 23, 23, 1},
%e A168290 {1, 59, 141, 59, 1},
%e A168290 {1, 135, 615, 615, 135, 1},
%e A168290 {1, 291, 2305, 4335, 2305, 291, 1},
%e A168290 {1, 607, 7971, 25415, 25415, 7971, 607, 1},
%e A168290 {1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1},
%e A168290 {1, 2519, 84191, 656039, 1722251, 1722251, 656039, 84191, 2519, 1},
%e A168290 {1, 5075, 264345, 3062055, 12001605, 18650247, 12001605, 3062055, 264345, 5075, 1}
%t A168290 w = 5; p[t_] = (1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%t A168290 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168290 Flatten[a]
%Y A168290 A046802
%K A168290 nonn,uned,new
%O A168290 0,5
%A A168290 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168018
%S A168018 1,2,1,3,1,5,2,1,7,1,11,3,2,1,15,1,22,5,2,1,30,3,1,42,7,2,1,56,1,77,11,
%T A168018 5,3,2,1
%N A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.
%C A168018 Positive values of triangle A168021.
%C A168018 Note that column 1 lists the numbers of partitions A000041(n).
%C A168018 Row n has a000005(n) terms.
%H A168018 O. E. Pol,<a href="www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the partitions of n, for n = 1 .. 9</a>
%e A168018 For example:
%e A168018 Consider the row 8: (22, 5, 2, 1). The divisor of 8 are 1, 2, 4, 8 (See A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22). Also, there are 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}. Also, there are 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}. Finally, there is 1 partition of 8 into parts divisible by 8. Then the row 8 is formed by 22, 5, 2, 1.
%e A168018 Triangle begins:
%e A168018 1;
%e A168018 2, 1;
%e A168018 3, 1;
%e A168018 5, 2, 1;
%e A168018 7, 1;
%e A168018 11, 3, 2, 1;
%e A168018 15, 1;
%e A168018 22, 5, 2, 1;
%e A168018 30, 3, 1;
%e A168018 42, 7, 2, 1;
%e A168018 56, 1;
%e A168018 77, 11, 5, 3, 2, 1;
%Y A168018 Row sums give A047968.
%Y A168018 Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.
%K A168018 more,nonn,tabf,new
%O A168018 1,2
%A A168018 Omar E. Pol (info(AT)polprimos.com), Nov 22 2009

%I A168289
%S A168289 1,1,1,1,6,1,1,19,19,1,1,48,114,48,1,1,109,494,494,109,1,1,234,1847,
%T A168289 3472,1847,234,1,1,487,6381,20339,20339,6381,487,1,1,996,21040,107028,
%U A168289 179506,107028,21040,996,1,1,2017,67360,524848,1377826,1377826,524848
%N A168289 Coefficients of the expansion of:w=4;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%C A168289 A umbral calculus expansion made from a Bezier extrapolation of the Pascal and A046802.
%C A168289 Row sums are:
%C A168289 {1, 2, 8, 40, 212, 1208, 7636, 54416, 437636, 3944104, 39453332,...}
%F A168289 w=4;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%e A168289 {1},
%e A168289 {1, 1},
%e A168289 {1, 6, 1},
%e A168289 {1, 19, 19, 1},
%e A168289 {1, 48, 114, 48, 1},
%e A168289 {1, 109, 494, 494, 109, 1},
%e A168289 {1, 234, 1847, 3472, 1847, 234, 1},
%e A168289 {1, 487, 6381, 20339, 20339, 6381, 487, 1},
%e A168289 {1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1},
%e A168289 {1, 2017, 67360, 524848, 1377826, 1377826, 524848, 67360, 2017, 1},
%e A168289 {1, 4062, 211485, 2449668, 9601326, 14920248, 9601326, 2449668, 211485, 4062, 1}
%t A168289 w = 4; p[t_] = (1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%t A168289 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168289 Flatten[a]
%Y A168289 A046802
%K A168289 nonn,uned,new
%O A168289 0,5
%A A168289 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168288
%S A168288 1,1,1,1,5,1,1,15,15,1,1,37,87,37,1,1,83,373,373,83,1,1,177,1389,2609,
%T A168288 1389,177,1,1,367,4791,15263,15263,4791,367,1,1,749,15787,80285,134647,
%U A168288 80285,15787,749,1,1,1515,50529,393657,1033401,1033401,393657,50529
%N A168288 Coefficients of the expansion of:w=3;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%C A168288 A umbral calculus expansion made from a Bezier extrapolation of the Pascal and A046802.
%C A168288 Row sums are:
%C A168288 {1, 2, 7, 32, 163, 914, 5743, 40844, 328291, 2958206, 29590255,...}
%F A168288 w=3;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%e A168288 {1},
%e A168288 {1, 1},
%e A168288 {1, 5, 1},
%e A168288 {1, 15, 15, 1},
%e A168288 {1, 37, 87, 37, 1},
%e A168288 {1, 83, 373, 373, 83, 1},
%e A168288 {1, 177, 1389, 2609, 1389, 177, 1},
%e A168288 {1, 367, 4791, 15263, 15263, 4791, 367, 1},
%e A168288 {1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1},
%e A168288 {1, 1515, 50529, 393657, 1033401, 1033401, 393657, 50529, 1515, 1},
%e A168288 {1, 3049, 158625, 1837281, 7201047, 11190249, 7201047, 1837281, 158625, 3049, 1}
%t A168288 w = 3; p[t_] = (1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%t A168288 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168288 Flatten[a]
%Y A168288 Cf. A046802
%K A168288 nonn,uned,new
%O A168288 0,5
%A A168288 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168287
%S A168287 1,1,1,1,4,1,1,11,11,1,1,26,60,26,1,1,57,252,252,57,1,1,120,931,1746,
%T A168287 931,120,1,1,247,3201,10187,10187,3201,247,1,1,502,10534,53542,89788,
%U A168287 53542,10534,502,1,1,1013,33698,262466,688976,688976,262466,33698,1013
%N A168287 Coefficients of the expansion of:w=2;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%C A168287 A sub-Eulerian number umbral calculus expansion made from a Bezier extrapolation of the Pascal and A046802.
%C A168287 Row sums are:
%C A168287 {1, 2, 6, 24, 114, 620, 3850, 27272, 218946, 1972308, 19727178,...}
%F A168287 w=2;p(t,x)=(1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%e A168287 {1},
%e A168287 {1, 1},
%e A168287 {1, 4, 1},
%e A168287 {1, 11, 11, 1},
%e A168287 {1, 26, 60, 26, 1},
%e A168287 {1, 57, 252, 252, 57, 1},
%e A168287 {1, 120, 931, 1746, 931, 120, 1},
%e A168287 {1, 247, 3201, 10187, 10187, 3201, 247, 1},
%e A168287 {1, 502, 10534, 53542, 89788, 53542, 10534, 502, 1},
%e A168287 {1, 1013, 33698, 262466, 688976, 688976, 262466, 33698, 1013, 1},
%e A168287 {1, 2036, 105765, 1224894, 4800768, 7460250, 4800768, 1224894, 105765, 2036, 1}
%t A168287 w = 2; p[t_] = (1 - w)Exp[t*(1 + x)] + w(1 - x)*(Exp[t])/(1 - x*Exp[t*(1 - x)])
%t A168287 a = Table[CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];
%t A168287 Flatten[a]
%Y A168287 Cf. A046802
%K A168287 nonn,uned,new
%O A168287 0,5
%A A168287 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 22 2009

%I A168286
%S A168286 2,8,8,14,14,20,20,26,26,32,32,38,38,44,44,50,50,56,56,62,62,68,68,74,
%T A168286 74,80,80,86,86,92,92,98,98,104,104,110,110,116,116,122,122,128,128,134,
%U A168286 134,140,140,146,146,152,152,158,158,164,164,170,170,176,176,182,182
%N A168286 a(n)=6*n-a(n-1)-2 (with a(1)=2)
%F A168286 a(n)=6*n-a(n-1)-2 (with a(1)=2)
%e A168286 For n=2, a(2)=6*2-2-2=8; n=3, a(3)=6*3-8-2=8; n=4, a(4)=6*4-8-2=14
%K A168286 nonn,new
%O A168286 1,1
%A A168286 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168285
%S A168285 1,3,7,11,14,19,25,29,32,37,43,51,52,57,65,72,75,76,79,83,99,104,105,
%T A168285 114,115,125,128,133,139,149,152,153,175,178,179,182,187,197,202,207,
%U A168285 212,213,221,231,244,247,248,251,269,278,279,287,299,304,307,312,319
%N A168285 nth nonprime prime minus nth nonprime.
%F A168285 a(n)= A000040(A018252(n))-A018252(n)=A007821(n)-A018252(n).
%K A168285 nonn,new
%O A168285 1,2
%A A168285 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168284
%S A168284 0,5,5,10,10,15,15,20,20,25,25,30,30,35,35,40,40,45,45,50,50,55,55,60,
%T A168284 60,65,65,70,70,75,75,80,80,85,85,90,90,95,95,100,100,105,105,110,110,
%U A168284 115,115,120,120,125,125,130,130,135,135,140,140,145,145,150,150,155
%N A168284 a(n)=5*n-a(n-1)-5 (with a(1)=0)
%F A168284 a(n)=5*n-a(n-1)-5 (with a(1)=0)
%e A168284 For n=2, a(2)=5*2-0-5=5; n=3, a(3)=5*3-5-5=5; n=4, a(4)=5*4-5-5=10
%K A168284 nonn,new
%O A168284 1,2
%A A168284 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168283
%S A168283 3,3,8,8,13,13,18,18,23,23,28,28,33,33,38,38,43,43,48,48,53,53,58,58,63,
%T A168283 63,68,68,73,73,78,78,83,83,88,88,93,93,98,98,103,103,108,108,113,113,
%U A168283 118,118,123,123,128,128,133,133,138,138,143,143,148,148,153,153,158
%N A168283 a(n)=5*n-a(n-1)-4 (with a(1)=3)
%F A168283 a(n)=5*n-a(n-1)-4 (with a(1)=3)
%e A168283 For n=2, a(2)=5*2-3-4=3; n=3, a(3)=5*3-4-3=8; n=4, a(4)=5*4-8-4=8
%K A168283 nonn,new
%O A168283 1,1
%A A168283 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168282
%S A168282 1,6,6,11,11,16,16,21,21,26,26,31,31,36,36,41,41,46,46,51,51,56,56,61,
%T A168282 61,66,66,71,71,76,76,81,81,86,86,91,91,96,96,101,101,106,106,111,111,
%U A168282 116,116,121,121,126,126,131,131,136,136,141,141,146,146,151,151,156
%N A168282 a(n)=5*n-a(n-1)-3 (with a(1)=1)
%F A168282 a(n)=5*n-a(n-1)-3 (with a(1)=1)
%e A168282 For n=2, a(2)=5*2-1-3=6; n=3, a(3)=5*3-6-3=6; n=4, a(4)=5*4-6-3=11
%K A168282 nonn,new
%O A168282 1,2
%A A168282 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168279
%S A168279 1,3,3,5,7,8,8,10,11,13,14,15,15,17,19,21,21,23,23,23,26,28,29,29,31,31,
%T A168279 33,33,33,37,38,39,39,41,43,45,45,46,46,47,49,52,52,52,53,55,59,60,60,
%U A168279 61,61,61,64,65,65,66,67,67,69,69,71,74,75,75,76,79,79,81,81,83,85,86
%N A168279 (n+1)-th prime nonprime minus (n+1)-th prime.
%F A168279 a(n)=A141468(A000040(n+1))-A000040(n+1)=A144570(n+1)-A000040(n+1).
%Y A168279 Cf. A000040(the primes), A141468(the positive nonprimes), A144570(the prime nonprimes).
%K A168279 nonn,new
%O A168279 1,2
%A A168279 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168281
%S A168281 2,2,2,2,8,2,2,8,8,2,2,8,18,8,2,2,8,18,18,8,2,2,8,18,32,18,8,2,2,8,18,
%T A168281 32,32,18,8,2,2,8,18,32,50,32,18,8,2,2,8,18,32,50,50,32,18,8,2
%N A168281 From extended periodic table of the elements. Take in A168208=1,2,2,1,2,2,2,3,2, symmetric consecutive terms which sum is A000027 with only A001105(n+1)=2,8,18,.
%C A168281 Sum by A000027 terms is 2,4,12,20,38, =2,(A099956 atomic numbers of alkaline earth metals) = 2*(A005993 alkane or paraffin numbers 1(6,n)) = Janet table first column from right to left. A137508=2*A106314(n+1).
%F A168281 a(n)=2,(A137508=2,2,2,8,2,) = 2*(A106314 for paraffin numbers).
%K A168281 nonn,uned,new
%O A168281 1,1
%A A168281 Paul Curtz (bpcrtz(AT)free.fr), Nov 22 2009

%I A168280
%S A168280 4,4,9,9,14,14,19,19,24,24,29,29,34,34,39,39,44,44,49,49,54,54,59,59,64,
%T A168280 64,69,69,74,74,79,79,84,84,89,89,94,94,99,99,104,104,109,109,114,114,
%U A168280 119,119,124,124,129,129,134,134,139,139,144,144,149,149,154,154,159
%N A168280 a(n)=5*n-a(n-1)-2 (with a(1)=4)
%F A168280 a(n)=5*n-a(n-1)-2 (with a(1)=4)
%e A168280 For n=2, a(2)=5*2-4-2=4; n=3, a(3)=5*3-4-2=9; n=4, a(4)=5*4-9-2=9
%K A168280 nonn,new
%O A168280 1,1
%A A168280 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168271
%S A168271 1,3,5,9,7,9,12,16,14,13,17,20,17,21,25,25,21,21,17,19,35,32,38,31,25,
%T A168271 31,33,33,39,47,33,30,47,49,39,39,37,43,44,44,43,41,38,48,58,49,47,35,
%U A168271 50,58,55,59,71,64,61,61,62,71,75,71,81,75,62,63,69,70,57,63,59,61,63
%N A168271 a(n) = prime(nonprime(n)) - nonprime(prime(n)).
%F A168271 a(n)=A007821(n)-A144570(n).
%Y A168271 Cf. A007821(the "nonprime primes"), A144570(the "prime nonprimes").
%K A168271 nonn,new
%O A168271 1,2
%A A168271 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168278
%S A168278 2,7,7,12,12,17,17,22,22,27,27,32,32,37,37,42,42,47,47,52,52,57,57,62,
%T A168278 62,67,67,72,72,77,77,82,82,87,87,92,92,97,97,102,102,107,107,112,112,
%U A168278 117,117,122,122,127,127,132,132,137,137,142,142,147,147,152,152,157
%N A168278 a(n)=5*n-a(n-1)-1 (with a(1)=2)
%e A168278 For n=2, a(2)=5*2-2-1=7; n=3, a(3)=5*3-7-1=7; n=4, a(4)=5*4-7-1=12
%K A168278 nonn,new
%O A168278 1,1
%A A168278 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168277
%S A168277 1,1,5,5,9,9,13,13,17,17,21,21,25,25,29,29,33,33,37,37,41,41,45,45,49,
%T A168277 49,53,53,57,57,61,61,65,65,69,69,73,73,77,77,81,81,85,85,89,89,93,93,
%U A168277 97,97,101,101,105,105,109,109,113,113,117,117,121,121,125,125,129,129
%N A168277 a(n)=4*n-a(n-1)-6 (with a(1)=1)
%e A168277 For n=2, a(2)=4*2-1-6=1; n=3, a(3)=4*3-1-6=5; n=4, a(4)=4*4-5-6=5
%K A168277 nonn,new
%O A168277 1,3
%A A168277 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168276
%S A168276 2,2,6,6,10,10,14,14,18,18,22,22,26,26,30,30,34,34,38,38,42,42,46,46,50,
%T A168276 50,54,54,58,58,62,62,66,66,70,70,74,74,78,78,82,82,86,86,90,90,94,94,
%U A168276 98,98,102,102,106,106,110,110,114,114,118,118,122,122,126,126,130,130
%N A168276 a(n)=4*n-a(n-1)-4 (with a(1)=2)
%e A168276 For n=2, a(2)=4*2-2-4=2; n=3, a(3)=4*3-2-4=6; n=4, a(4)=4*4-6-4=6
%Y A168276 Cf. A039722
%K A168276 nonn,new
%O A168276 1,1
%A A168276 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168275
%S A168275 3,4,7,6,7,8,11,11,11,12,12,14,15,15,17
%N A168275 Number of letters in n-th prime (in Latin)
%H A168275 Arndt Bruenner, <a href="http://www.arndt-bruenner.de/mathe/scripts/numeraliatab.htm">Table of Latin numerals</a>
%e A168275 a(1) = 3, because the first prime number is 2 and it contains 3 letters in Latin.
%K A168275 hard,nonn,word,new
%O A168275 1,1
%A A168275 Ivan Panchenko (panchenko.ivan1(AT)gmail.com), Nov 22 2009

%I A168272
%S A168272 1,2,2,2,8,12,22,24,26,56,66,86,106,112,122,144,176,180,224,240,236,264,
%T A168272 292,312,358,384,396,414,418,424,512,540,550,570,630,644,680,716,734,
%U A168272 768,794,816,872,878,894,906,984,1092,1096,1100,1122,1140,1150,1242
%N A168272 Abs(prime(prime(n))-prime(nonprime(n))).
%F A168272 a(n)=abs(A006450(n)-A007821(n)).
%e A168272 a(1)=abs(3-2)=1, a(2)=abs(5-7)=2, a(3)=abs(11-13)=2, a(4)=abs(17-19)=2, a(5)=abs(31-23)=8.
%Y A168272 Cf. A000040(nth prime), A006450((nth prime)th prime), A007821((nth nonprime)th prime), A018252(nth nonprime).
%K A168272 nonn,new
%O A168272 1,2
%A A168272 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168273
%S A168273 0,4,4,8,8,12,12,16,16,20,20,24,24,28,28,32,32,36,36,40,40,44,44,48,48,
%T A168273 52,52,56,56,60,60,64,64,68,68,72,72,76,76,80,80,84,84,88,88,92,92,96,
%U A168273 96,100,100,104,104,108,108,112,112,116,116,120,120,124,124,128,128,132
%N A168273 a(n)=4*n-a(n-1)-4 (with a(1)=0)
%e A168273 For n=2, a(2)=4*2-0-4=4; n=3, a(3)=4*3-4-4=4; n=4, a(4)=4*4-4-4=8
%K A168273 nonn,new
%O A168273 1,2
%A A168273 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168274
%S A168274 3,31,37,43,73,109,163,211,241,313,337,409,499,673,739,757,907,1033,
%T A168274 1039,1069,1237,1327,1447,1483,1489,1597,1663,1741,1777,1933,2083,2143,
%U A168274 2251,2437,2683,2797,3001,3181,3307,3319,3463,3739,3793,4051,4153,4201
%N A168274 Primes p in (A168219) with q=1 + 10*p^3 (A168147) a prime too
%C A168274 It is conjectured that sequence is infinite
%D A168274 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A168274 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D A168274 Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996
%e A168274 (1) 1+10*3^3=271=prime(58) gives 3=prime(2)=a(1)
%e A168274 (2) 1+10*31^3=297911=prime(25840) gives 31=prime(11)=a(2)
%e A168274 (3) 1+10*313^3=306642971=prime(16592480) gives 313=prime(65)=a(10)
%e A168274 (4) 1+10*1327^3=23367527831=prime(1023707296) gives 1327=prime(217)=a(22)
%Y A168274 Cf. A000040 The prime numbers
%Y A168274 Cf. A168147
%Y A168274 Cf. A168219 Naturals n for which 1 + 10*n^3 is prime
%Y A168274 Cf. A167535 Concatenation of two square numbers which give a prime
%K A168274 nonn,new
%O A168274 1,1
%A A168274 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 22 2009

%I A168270
%S A168270 2,6,23,30,37,42,47,53,67,79,83,98,97,102,113
%N A168270 Square-free numbers which are also single or isolated numbers.
%Y A168270 Cf. A005117(square-free numbers), A167706(single or isolated numbers), A168252.
%K A168270 nonn,new
%O A168270 1,1
%A A168270 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009

%I A168269
%S A168269 3,3,7,7,11,11,15,15,19,19,23,23,27,27,31,31,35,35,39,39,43,43,47,47,51,
%T A168269 51,55,55,59,59,63,63,67,67,71,71,75,75,79,79,83,83,87,87,91,91,95,95,
%U A168269 99,99,103,103,107,107,111,111,115,115,119,119,123,123,127,127,131,131
%N A168269 a(n)=4*n-a(n-1)-2 (with a(1)=3)
%e A168269 For n=2, a(2)=4*2-3-2=3; n=3, a(3)=4*3-3-2=7; n=4, a(4)=4*4-7-2=7
%K A168269 nonn,new
%O A168269 1,1
%A A168269 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168268
%S A168268 0,0,3,3,6,6,9,9,12,12,15,15,18,18,21,21,24,24,27,27,30,30,33,33,36,36,
%T A168268 39,39,42,42,45,45,48,48,51,51,54,54,57,57,60,60,63,63,66,66,69,69,72,
%U A168268 72,75,75,78,78,81,81,84,84,87,87,90,90,93,93,96,96,99,99,102,102,105
%N A168268 a(n)=3*n-a(n-1)-6 (with a(1)=0)
%e A168268 For n=2, a(2)=3*2-0-6=0; n=3, a(3)=3*3-0-6=3; n=4, a(4)=3*4-3-6=3
%K A168268 nonn,new
%O A168268 1,3
%A A168268 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009

%I A168160
%S A168160 0,2,2,7,8,9,9,19,21,23,24,26,27,28,28,47,50,53,55,58,60,62,63,66,68,70,
%T A168160 71,73,74,75,75,111,115,119,122,126,129,132,134,138,141,144,146,149,151,
%U A168160 153,154,158,161,164,166,169,171,173,174,177,179,181,182,184,185,186
%N A168160 Number of 0's in the matrix whose lines are the binary expansion of the numbers 1,...,n.
%C A168160 The matrix is to be taken of minimal size, i.e., have n lines and the number of columns needed to write n in base 2 in the last line, A070939(n). Otherwise said, there is no zero column.
%C A168160 The number of zeros in the last line of the matrix is given by A023416(n).
%C A168160 One has a(n)=a(n-1) iff n = 2^k-1 for some k.
%F A168160 A168160(n)=n*A070939(n)-A000788(n).
%e A168160 a(4)=7 is the number of zeros in the matrix
%e A168160 [001] /* = 1 in binary */
%e A168160 [010] /* = 2 in binary */
%e A168160 [011] /* = 3 in binary */
%e A168160 [100] /* = 4 in binary */
%o A168160 (PARI) A168160(n)=n*#binary(n)-sum(i=1,n,norml2(binary(i)))
%Y A168160 Cf. A059015.
%K A168160 base,nonn,new
%O A168160 1,2
%A A168160 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 22 2009

%I A168261
%S A168261 1,1,1,0,0,2,1,1,0,2,0,0,0,0,4,0,0,2,0,0,4,0,0,0,0,0,0,6,1,1,0,2,0,0,0,
%T A168261 6,0,0,0,0,0,0,0,0,10,0,0,0,0,4,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,14,0,0,2,
%U A168261 0,0,4,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,20
%N A168261 Triangle by rows, A115361 * the diagonalized variant of A018819.
%C A168261 Row sums = A018819 starting with offset 1; (1, 2, 2, 4, 4, 6, 6, 10, 10,...).
%C A168261 = the eigensequence of triangle A115361.
%C A168261 Rightmost diagonal = A018819
%C A168261 Sum of n-th row terms = rightmost term of next row.
%F A168261 Let M = triangle A115361, and Q = the diagonalized variant of A018819 such
%F A168261 that (1, 1, 2, 2, 4, 4, 6, 6,...) = rightmost diagonal with the rest zeros.
%F A168261 Triangle A168261 = M*Q as infinite lower triangular matrices.
%e A168261 First few rows of the triangle =
%e A168261 1;
%e A168261 1, 1;
%e A168261 0, 0, 2;
%e A168261 1, 1, 0, 2;
%e A168261 0, 0, 0, 0, 4;
%e A168261 0, 0, 2, 0, 0, 4;
%e A168261 0, 0, 0, 0, 0 0, 6;
%e A168261 1, 1, 0, 2, 0, 0, 0, 6;
%e A168261 0, 0, 0, 0, 0, 0, 0, 0, 10;
%e A168261 0, 0, 0, 0, 4, 0, 0, 0, 0, 10;
%e A168261 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14;
%e A168261 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 14;
%e A168261 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20;
%e A168261 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 20;
%e A168261 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26;
%e A168261 1, 1, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 26;
%e A168261 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36;
%e A168261 ...
%Y A168261 Cf. A115361, A018819
%K A168261 nonn,tabl,new
%O A168261 1,6
%A A168261 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009

%I A168260
%S A168260 1,1,1,2,2,2,2,2,4,6,3,3,6,12,14,3,3,6,18,28,38,4,4,8,24,42,76,96,4,4,8,
%T A168260 24,56,114,192,254,5,5,10,30,70,152,288,508,656,5,5,10,30,70,190,384,
%U A168260 762,1312,1724,6,6,12,36,84,228,480,1016
%N A168260 Triangle by rows, A168258 * the diagonalized variant of A168359
%C A168260 Row sums = A168259: (1, 2, 6, 14, 38, 96,...).
%C A168260 Sum of n-th row terms = rightmost term of next row.
%C A168260 Row sum ratios tend (conjecture) to phi^2, 2.6180339...(Cf. A168259).
%F A168260 Let M = triangle A168258 and Q = the diagonalized variant of M's eigensequence
%F A168260 such that Q's rightmost diagonal = A168259 prefaced with a 1: (1, 1, 2, 6,...).
%F A168260 and other terms = 0.
%F A168260 Triangle A168260 = M * Q as infinite lower triangular matrices.
%e A168260 First few rows of the triangle =
%e A168260 1
%e A168260 1, 1;
%e A168260 2, 2, 2;
%e A168260 2, 2, 4, 6;
%e A168260 3, 3, 6, 12, 14;
%e A168260 3, 3, 6, 18, 28, 38;
%e A168260 4, 4, 8, 24, 42, 76, 96;
%e A168260 4, 4, 8, 24, 56, 114, 192, 254;
%e A168260 5, 5, 10, 30, 70, 152, 288, 508, 656;
%e A168260 5, 5, 10, 30, 70, 190, 384, 762, 1312, 1724;
%e A168260 6, 6, 12, 36, 84, 228, 480, 1016, 1968, 3448, 4492;
%e A168260 6, 6, 12, 36, 84, 228, 576, 1270, 2624, 5172, 8984, 11776;
%e A168260 7, 7, 14, 42, 98, 266, 672, 1524, 3284, 6896, 13476, 23552, 30774;
%e A168260 7, 7, 14, 42, 98, 266, 672, 1778, 3936, 8620, 17968, 35328, 61548, 80608;
%e A168260 8, 8, 16, 48, 112, 304, 768, 2032, 5248, 12068, 26952, 58880, 123096, 241824;
%e A168260 ...
%Y A168260 Cf. A168258, A168259
%K A168260 nonn,tabl,new
%O A168260 1,4
%A A168260 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009

%I A168259
%S A168259 1,2,6,14,38,96,254,656,1724,4492,11776,30774,80608,210892,552226,
%T A168259 1445374,3784308,9906482,25936206
%N A168259 Eigensequence of triangle A168258. Triangle A168258 * the diagonalized variant of A168259 = triangle A168270 having row sums = A168259.
%C A168259 a(n)/a(n-1) apparently tends to phi^2. a(19)/a(18) = 2.618104...
%F A168259 Eigensequence of triangle A168258, derived from the following operation:
%F A168259 Shift down triangle A168258, so that rows begin [1; 1; 1,1; 2,2,1;...] =
%F A168259 triangle M. Then take Lim_{n->inf.} Q^n; resulting in a left-shifted vector.
%F A168259 Delete the first 1, getting (1, 2, 6, 14, 38, 96,...) = A168259. . Q^n} Q tri.
%Y A168259 Cf. A168258, A168260
%K A168259 eigen,nonn,new
%O A168259 1,2
%A A168259 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009

%I A168158
%S A168158 7,10,11,12,15,18,29,35,45,49,51,55,58,65,80,82,84,89,93,95,114,119,128,
%T A168158 130,140,142,149,155,157,160,166,171,173,175,191,192,209,210,218,235,
%U A168158 240,251,262,263,269,280,305,315,321,324,328,337,341,344,345,363,372
%N A168158 Indices n for which A168157(n) is prime.
%C A168158 If n=primepi(2^k-1) is in this sequence, then A158671(k) is prime. This happens for k=5, 6, 18,... corresponding to n=11, 18, 23000,... (cf. link).
%H A168158 C. Caldwell, G. L. Honaker, Eds.: <a href="http://primes.utm.edu/curios/page.php?short=18&submitter=Post">Prime curio!: 18</a> by J. vos Post.
%o A168158 (PARI) s=0; for(n=1,999, isprime(n*#(b=binary(prime(n)))-s+=norml2(b)) & print1(n", "))
%K A168158 nonn,new
%O A168158 1,1
%A A168158 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 21 2009

%I A168258
%S A168258 1,1,1,2,2,1,2,2,2,1,3,3,3,2,1,3,3,3,3,2,1,4,4,4,4,3,2,1,4,4,4,4,4,3,2,
%T A168258 1,5,5,5,5,5,5,4,3,2,1,5,5,5,5,5,5,4,3,2,1,6,6,6,6,6,6,5,4,3,2,1,6,6,6,
%U A168258 6,6,6,6,5,4,3,2,1
%N A168258 Triangle by rows, A097806 * A111967 as infinite lower triangular matrices.
%C A168258 Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22,...).
%C A168258 Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656,...).
%F A168258 Triangle by Rows, A097806 * A111967; A097806 = the signed pairwise operator
%F A168258 (1's in the main diagonal, -1's in the adjacent diagonal and the rest zeros.)
%e A168258 First few rows of the triangle =
%e A168258 1;
%e A168258 1, 1;
%e A168258 2, 2, 1;
%e A168258 2, 2, 2, 1;
%e A168258 3, 3, 3, 2, 1;
%e A168258 3, 3, 3, 3, 2, 1;
%e A168258 4, 4, 4, 4, 3, 2, 1;
%e A168258 4, 4, 4, 4, 4, 3, 2, 1;
%e A168258 5, 5, 5, 5, 5, 4, 3, 2, 1;
%e A168258 5, 5, 5, 5, 5, 5, 4, 3, 2, 1;
%e A168258 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
%e A168258 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
%e A168258 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
%e A168258 7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
%e A168258 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;
%e A168258 ...
%Y A168258 Cf. A001318, A097806, A111967, A168259
%K A168258 nonn,tabl,new
%O A168258 1,4
%A A168258 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009

%I A168257
%S A168257 0,1,9,784
%N A168257 Numbers n with property that 4+2*n+3*n^3 is a square.
%C A168257 4+2*n+3*n^3 = m^2 with n = {0,1,9,784} = {0,1,3,28}^2, and m = 2,3,47,38022.
%K A168257 fini,full,nonn,new
%O A168257 0,3
%A A168257 Zak Seidov (zakseidov(AT)yahoo.com), Nov 21 2009

%I A168157
%S A168157 1,1,4,4,9,10,19,21,22,23,23,37,40,42,43,45,46,47,69,72,76,78,81,84,88,
%T A168157 91,93,95,97,100,100,136,141,145,149,152,155,159,162,165,168,171,172,
%U A168157 177,181,184,187,188,191,194,197,198,201,202,263,268,273,277,282,287
%N A168157 Number of 0's in the matrix whose lines are the binary expansion of the first n primes.
%C A168157 The matrix is to be taken of minimal size, i.e., have n lines and the number of columns needed to write the n-th prime in the last line, A035100(n). Otherwise said, there is no zero column except for n=1 (prime(1) = 2 = 10[2] in binary).
%C A168157 The number of zeros in the last line of the matrix is given by A035103(n).
%C A168157 One has a(n)=a(n-1) iff n = A059305(k) for some k, i.e. prime(n) is a Mersenne prime A000668(k) = A000225(A000043(k)).
%C A168157 If prime(n)=2^2^k+1 is a Fermat prime (A019434), n>2, then one has a(n)=a(n-1)+n-1+2^k-1.
%C A168157 More generally, the "big jumps" a(n+1) > a(n)+n happen whenever a column is added, i.e. when prime(n) = A014234(k) <=> prime(n+1) = A104080(k) for some k,n>1.
%F A168157 a(n)=n*A035100(n)-A095375(n).
%e A168157 a(4)=4 is the number of zeros in the matrix [010] /* = 2 in binary */ [011] /* = 3 in binary */ [101] /* = 5 in binary */ [111] /* = 7 in binary */
%o A168157 (PARI) A168157(n)=n*#binary(prime(n))-sum(i=1,n,norml2(binary(prime(i))))
%K A168157 base,nonn,new
%O A168157 1,3
%A A168157 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 21 2009

%I A166887
%S A166887 1,3,18,157,1812,25989,445255,8865333,201058614,5114874693,144207579708,
%T A166887 4462151144553,150316762118466,5475746846833734,214463847533104125,
%U A166887 8986421286160678944,401112805593137715609,18999650382886046745879
%N A166887 Column 3 of triangle A166884.
%C A166887 Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.
%o A166887 (PARI) {a(n)=local(F=x, M, N, P, m=n+2); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}
%Y A166887 Cf. A166880, A166884, A166885, A166886.
%K A166887 nonn,new
%O A166887 1,2
%A A166887 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A166886
%S A166886 1,2,9,62,593,7266,108720,1922166,39212154,906623004,23429034168,
%T A166886 669203550906,20935080981744,711872134399868,26142553369667634,
%U A166886 1031146768716808794,43475757877044427198,1951261759908828697902
%N A166886 Column 2 of triangle A166884.
%C A166886 Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.
%o A166886 (PARI) {a(n)=local(F=x, M, N, P, m=n+1); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+2, 2]}
%Y A166886 Cf. A166880, A166884, A166885, A166887.
%K A166886 nonn,new
%O A166886 1,2
%A A166886 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A166885
%S A166885 1,1,3,15,114,1159,14838,229401,4159662,86580636,2034850425,53303009286,
%T A166885 1539990513588,48648616439496,1668228105283302,61715049142446537,
%U A166885 2450018515737072792,103892256368706869356,4686744256645813560957
%N A166885 Column 1 of triangle A166884.
%C A166885 Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.
%o A166885 (PARI) {a(n)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, 1]}
%Y A166885 Cf. A166880, A166884, A166886, A166887.
%K A166885 nonn,new
%O A166885 1,3
%A A166885 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A166884
%S A166884 1,1,1,3,2,1,15,9,3,1,114,62,18,4,1,1159,593,157,30,5,1,14838,7266,1812,
%T A166884 316,45,6,1,229401,108720,25989,4271,555,63,7,1,4159662,1922166,445255,
%U A166884 70180,8595,890,84,8,1,86580636,39212154,8865333,1354750,159171,15534
%N A166884 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).
%e A166884 This triangle begins:
%e A166884 1;
%e A166884 1,1;
%e A166884 3,2,1;
%e A166884 15,9,3,1;
%e A166884 114,62,18,4,1;
%e A166884 1159,593,157,30,5,1;
%e A166884 14838,7266,1812,316,45,6,1;
%e A166884 229401,108720,25989,4271,555,63,7,1;
%e A166884 4159662,1922166,445255,70180,8595,890,84,8,1;
%e A166884 86580636,39212154,8865333,1354750,159171,15534,1337,108,9,1;
%e A166884 2034850425,906623004,201058614,30000676,3418245,320070,25963,1912,135,10,1;
%e A166884 53303009286,23429034168,5114874693,748896765,83336385,7568355,589057,40882,2631,165,11,1; ...
%e A166884 Triangle A166880 of coefficients in iterations of x+x^2+x^3 begins:
%e A166884 1;
%e A166884 1,1,1;
%e A166884 1,2,4,6,8,8,6,3,1;
%e A166884 1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,...;
%e A166884 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
%e A166884 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
%e A166884 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
%e A166884 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
%e A166884 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...; ...
%e A166884 in which this triangle transforms diagonals in A166880 into each other.
%e A166884 The initial diagonals in triangle A166880 begin:
%e A166884 A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
%e A166884 A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
%e A166884 A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
%e A166884 so that, if we treat the diagonals as column vectors, we have:
%e A166884 A166884 * A166881 = A166882,
%e A166884 A166884 * A166882 = A166883.
%o A166884 (PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%Y A166884 Cf. A166880, columns: A166885, A166886, A166887.
%K A166884 nonn,tabl,new
%O A166884 1,4
%A A166884 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A168159
%S A168159 1,1,1,9,7,49,33,169,7,7,207,237,91,313,261,273,79,49,2901,51,441,193,9,
%T A168159 531,289,1141,67,909,331,753,2613,657,49,4459,603,1531,849,2049,259,649,
%U A168159 2119,1483,63,6747,519,3133,937,1159,1999,6921,2949,613,4137,1977,31
%N A168159 Distance of the least reversible n-digit prime from 10^(n-1)
%C A168159 A (much) more compact form of A114018 (cf. formula). Since this sequence and A114018 refer to "reversible primes" (A007500), while A122490 seems to use "emirps" (A006567), a(n+1) differs from A122490(n) iff 10^n+1 is prime <=> a(n+1)=1 <=> A114018(n)=10^n+1.
%F A168159 a(n)=A114018(n)-10^(n-1)
%o A168159 (PARI) for(x=1,1e99, until( isprime(x=nextprime(x+1)) & isprime(eval(concat(vecextract(Vec(Str(x)),"-1..1")))),);print1(x-10^ (#Str(x)-1),", "); x=10^#Str(x)-1)
%K A168159 base,nonn,new
%O A168159 1,4
%A A168159 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 21 2009

%I A168256
%S A168256 1,1,1,2,2,2,5,5,5,5,14,14,14,14,14,42,42,42,42,42,42,132,132,132,132,
%T A168256 132,132,132
%N A168256 n*Catalan number(n)triangle
%C A168256 Rows sum up to A000984
%e A168256 Triangle begins:
%e A168256 1
%e A168256 1,1
%e A168256 2,2,2
%e A168256 5,5,5,5
%Y A168256 A000108, A000984
%K A168256 nonn,tabl,new
%O A168256 1,4
%A A168256 Mark Dols (Markdols99(AT)yahoo.com), Nov 21 2009

%I A166880
%S A166880 1,1,1,1,1,2,4,6,8,8,6,3,1,1,3,9,24,60,138,294,579,1053,1767,2739,3924,
%T A166880 5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1,1,4,16,
%U A166880 60,216,744,2460,7818,23910,70446,200160,549006,1455132,3730846,9262712
%N A166880 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of (x+x^2+x^3).
%e A166880 Triangle begins:
%e A166880 1;
%e A166880 1,1,1;
%e A166880 1,2,4,6,8,8,6,3,1;
%e A166880 1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1;
%e A166880 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
%e A166880 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
%e A166880 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
%e A166880 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
%e A166880 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...;
%e A166880 1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;
%e A166880 1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;
%e A166880 1,11,121,1320,14300,153890,1645710,17494455,184915225,...;
%e A166880 1,12,144,1716,20328,239448,2805396,32700558,379309986,...;
%e A166880 1,13,169,2184,28080,359268,4575324,58009614,732380298,...;
%e A166880 1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;
%e A166880 1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;
%e A166880 1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;
%e A166880 1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;
%e A166880 1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;
%e A166880 ...
%e A166880 The initial diagonals in this triangle begin:
%e A166880 A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
%e A166880 A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
%e A166880 A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
%e A166880 The diagonals are transformed one into the other by
%e A166880 triangle A166884, which begins:
%e A166880 1;
%e A166880 1,1;
%e A166880 3,2,1;
%e A166880 15,9,3,1;
%e A166880 114,62,18,4,1;
%e A166880 1159,593,157,30,5,1;
%e A166880 14838,7266,1812,316,45,6,1;
%e A166880 229401,108720,25989,4271,555,63,7,1;
%e A166880 4159662,1922166,445255,70180,8595,890,84,8,1; ...
%o A166880 (PARI) {T(n, k)=local(F=x+x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
%Y A166880 Cf. diagonals: A166881, A166882, A166883, related triangle: A166884.
%Y A166880 Cf. row sums: A166999, variant: A122888.
%K A166880 nonn,tabf,new
%O A166880 0,6
%A A166880 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A166999
%S A166999 1,3,39,60879,225636660844959,
%T A166999 11487591726386681145142587842614325062822719
%N A166999 a(n) = a(n-1) + a(n-1)^2 + a(n-1)^3 for n>0 with a(0)=1.
%F A166999 Row sums of triangle A166880, which lists the coefficients in the iterations of x+x^2+x^3.
%o A166999 (PARI) a(n)=if(n==0,1,a(n-1)+a(n-1)^2+a(n-1)^3)
%Y A166999 Cf. A166880.
%K A166999 nonn,new
%O A166999 0,2
%A A166999 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 21 2009

%I A168255
%S A168255 1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,
%T A168255 6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,
%U A168255 9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10
%N A168255 n appears n-th nonprime number times.
%C A168255 n appears A018252(n) times.
%K A168255 nonn,new
%O A168255 1,2
%A A168255 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 21 2009

%I A168254
%S A168254 0,1,28,103,403,613,658,1018,1138,1318,1678,2023,2188,2383,2653,3418,
%T A168254 3853,3958,4333,4393,4828,5278,5908,6163,8218,8428,8683,8848,9028,9043,
%U A168254 9478,9868,10003,10243,10978,11518,11533,12133,12493,13603,13633,14593
%N A168254 Numbers n with property that n^3+n^2+{3,5} are twin primes.
%C A168254 All positive terms == 1 mod 3.
%t A168254 Union[Table[If[PrimeQ[n^3+n^2+3]&&PrimeQ[n^3+n^2+5],n,0],{n,1,40000,3}]]
%K A168254 nonn,new
%O A168254 0,3
%A A168254 Zak Seidov (zakseidov(AT)yahoo.com), Nov 21 2009

%I A168253
%S A168253 1,1,4,4,7,7,10,10,13,13,16,16,19,19,22,22,25,25,28,28,31,31,34,34,37,
%T A168253 37,40,40,43,43,46,46,49,49,52,52,55,55,58,58,61,61,64,64,67,67,70,70,
%U A168253 73,73,76,76,79,79,82,82,85,85,88,88,91,91,94,94,97,97,100,100,103,103
%N A168253 a(n)=3*n-a(n-1)-4 (with a(1)=1)
%e A168253 For n=2, a(2)=3*2-1-4=1; n=3, a(3)=3*3-1-4=4; n=4, a(4)=3*4-4-4=4
%K A168253 nonn,new
%O A168253 1,3
%A A168253 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009

%I A161321
%S A161321 0,9,1,6,0,7,9,7,8,3,0,9,9,6,1,6,0,4,2,5,6,7,3,2,8,2,9,1,5,6,1,6,1,7,
%T A161321 0,4,8,4,1,5,5,0,1,2,3,0,7,9,4,3,4,0,3,2,2,8,7,9,7,1,9,6,6,9,1,4,2,
%U A161321 8,2,2,4,5,9,1,0,5,6,5,3,0,3,6,7,6,5,7,5,2,5,2,7,1,8,3,1,0,9,1,7,8
%N A161321 Decimal expansion of (sqrt(35)-5)/10.
%D A161321 D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 321.
%e A161321 .091607978309961604256732829156161704841550123079434032287971...
%Y A161321 Cf. A090656, A010490.
%K A161321 nonn,cons,new
%O A161321 0,1
%A A161321 N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2009

%I A147313
%S A147313 1,6,5,8,3,1,2,3,9,5,1,7,7,6,9,9,9,2,4,5,5,7,4,6,6,3,6,8,3,3,5,3,4,
%T A147313 3,3,4,1,9,6,3,5,4,4,2,7,2,7,9,4,6,7,6,7,9,8,5,2,9,3,4,1,0,7,3,0,5,
%U A147313 8,2,4,2,3,2,1,3,0,4,5,2,1,9,2,3,3,5,4,4,2,1,6,9,9,5,6,4,1,4,5,3,2
%N A147313 Decimal expansion of sqrt(11)/2.
%D A147313 D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 308.
%e A147313 1.6583123951776999245574663683353433419635442727946767985293...
%K A147313 nonn,cons,new
%O A147313 1,2
%A A147313 N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2009

%I A161222
%S A161222 1,8,30,120,618,3536,22668,151848,1054986,7472984,53737896,390582648,2863716060,
%T A161222 21145502960,157076310324,1172820793824,8796118712586,66229473393728,500400163666188,
%U A161222 3792505486235544,28823039252629512,219604100410657136,1676976747053723292
%N A161222 Consider necklaces with n beads, each of one of four colors (say C1, C2, C3, C4), where the n segments of cord between the beads are each colored red or green; a(n) is the number of different necklaces under the action of the dihedral group D_{2n}.
%C A161222 If the group is changed to C_n we get A054627.
%F A161222 For formula see Maple code.
%p A161222 with(numtheory); f:=proc(n) local t1,d,m;
%p A161222 if n mod 2 = 0 then m:=n/2; t1:=3*2^(3*m);
%p A161222 else m:=(n-1)/2; t1:=2^(3*m+3); fi;
%p A161222 (1/2)*( (1/n) * add( phi(d)*2^(3*n/d), d in divisors(n)) + t1 );
%p A161222 end; # this assumes n>0
%Y A161222 Cf. A161221, A054627.
%K A161222 nonn,new
%O A161222 0,2
%A A161222 H. O. Pollak (hpollak(AT)adsight.com) and N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A161221
%S A161221 1,4,9,20,51,136,414,1300,4371,15084,53508,192700,703346,2589304,9603954,
%T A161221 35824240,134285331,505421344,1909144014,7234153420,27488865564,104717491064,
%U A161221 399826699734,1529763696820,5864079144466,22518031691368,86607753541164
%N A161221 Consider necklaces with n beads, each black or white, where the n segments of cord between the beads are each colored red or green; a(n) is the number of different necklaces under the action of the dihedral group D_{2n}.
%C A161221 If the group is changed to C_n we get A001868.
%F A161221 For n>0, a(n) = (1/2)*( (1/n) * Sum_{d|n} (phi(n/d)*2^(2*d)) + 2^(n+1) ).
%e A161221 a(4) = 51: the following table shows the number of such necklaces with b black beads, 4-b white beads, r red chord segments and 4-r green chord segments. The sum of the numbers is 51.
%e A161221 b\r 0 1 2 3 4
%e A161221 -------------
%e A161221 0 | 1 1 2 1 1
%e A161221 1 | 1 2 4 2 1
%e A161221 2 | 2 4 7 4 2
%e A161221 3 | 1 2 4 2 1
%e A161221 4 | 1 1 2 1 |
%p A161221 with(numtheory); f:= n-> (1/2)*( (1/n) * add( phi(n/d)*2^(2*d), d in divisors(n)) + 2^(n+1) ); # this assumes n>0
%Y A161221 Cf. A000029, A000031, A001868, A161222.
%K A161221 nonn,new
%O A161221 0,2
%A A161221 H. O. Pollak (hpollak(AT)adsight.com) and N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A168215
%S A168215 27346,85910,1916478335,6020794258,225843133696748,709507129685401
%N A168215 Numbers H(n) where H(2*n)/H(2*n-1) is composed with n times 0-9 and it's the colsest number to pi
%C A168215 H(3),H(4) was provided by medie2005,mathe,zgg__ H(5),H(6) was provided by mathe Ref: http://topic.csdn.net/u/20070825/21/2e2f6348-b0f8-46cb-a52b-582af245141a.html
%e A168215 H(3)=1916478335 H(4)=6020794258 Means H(4)/H(3) and=3.1415926535897939070623514249119 is the closest numer to pi.H(4)& H(3) were composed with 2 times 0-9 digits
%e A168215 #include <stdio.h> #include <time.h> #include <math.h> #include <assert.h> typedef struct tag_int6{ unsigned x[6]; }int6; #define NUMS 3 long long d, u; long long max_u; int delta; time_t start_time; char strd[5*NUMS+1]; char stru[5*NUMS+2]; double err=1.0; double twotimes64; long long best_d, best_u; int6 PI; int6 N_PI[5*NUMS+1]; int6 NA1_PI[5*NUMS+1]; //dst=10*src void time10(int6 *dst, int6 *src) { long long x0=src->x[0]; long long x1=src->x[1];
%o A168215 (Other) Provided by mathe in floor 92# of webpage: http://topic.csdn.net/u/20070825/21/2e2f6348-b0f8-46cb-a52b-582af245141a.html
%K A168215 hard,nice,nonn,new
%O A168215 2,1
%A A168215 Zhining Yang (northwolves(AT)163.com), Nov 20 2009

%I A168226
%S A168226 1890,2520,3150,3780,4410,5040,5670,6300,6720,6930,7350,7380,7560,7770,
%T A168226 7980,8010,8190,8370,8400,8610,8640,8820,9000,9030,9240,9270,9360,9450,
%U A168226 9630,9660,9870,9900,9990,10080,10260,10290,10500,10530,10620,10710
%N A168226 Even numbers which are the sum of two odd abundant numbers.
%C A168226 Every even number >= 3706141025766237065507279802221127212928 is the sum of two odd abundant numbers. The largest even number which does not appear in this sequence is unknown.
%e A168226 945 is the smallest odd abundant number, so 945 + 945 = 1890 is the first term in the sequence.
%Y A168226 Cf. A005231, A048260.
%K A168226 nonn,new
%O A168226 1,1
%A A168226 William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Nov 20 2009

%I A168248
%S A168248 1159,2217,3463,4161,5713,5911,6037,6451,6699,7333,9403,10249,10447,
%T A168248 10483,11019,11347,11383,13039,13503,14173,15127,15879,18489,19569,
%U A168248 20307,20563,21333,22921,23713,24613,25009,26877,27111,30207,30229
%N A168248 a(n) = A168174(n)-10^12.
%Y A168248 Cf. A168174 Emirps with emirp number and sum of digits, A006567 Emirps.
%K A168248 base,nonn,new
%O A168248 1,1
%A A168248 Zak Seidov (zakseidov(AT)yahoo.com), Nov 21 2009

%I A168240
%S A168240 21,67,139,237,361,511,687,889
%N A168240 In continuation of the recent sequence (cf A168235) the present sequence is one of quotients generated by (f(x + n*f(x))/f(x) when f(x) = x^2 + x +1 ( x =3 and n belongs to N).
%e A168240 f(x)= 13 when x =3. Hence f(x + f(x)/f(x) = 21.
%p A168240 {p(n) = ((3 + 13*n)^2 + 3 + 13*n + 1)/13}; for ( n= 1,10, print P(n))).
%Y A168240 Cf. A168235, A165806, A165808 and A165809
%K A168240 nonn,uned,new
%O A168240 1,1
%A A168240 A. K. Devaraj (dkandadai(AT)gmail.com), Nov 21 2009

%I A168250
%S A168250 17,13,17,13,17,17,17,17,31,17,17,17,17,17,13,17,17,17,13,17,17,31,31,
%T A168250 31,13,17,13,17,17,17,17,31,13,13,17,13,17,17,13,17,31,31,31,17,17,31,
%U A168250 13,17,31,17,17,13,37,31,17,31,17,31,31,13,31,13,17,17,31,37,31,17,31
%N A168250 a(n) = sum of digits of A168174(n).
%Y A168250 Cf. A006567, A168174, A168248.
%K A168250 base,nonn,new
%O A168250 1,1
%A A168250 Zak Seidov (zakseidov(AT)yahoo.com), Nov 21 2009

%I A168211
%S A168211 0,15,7,22,14,29,21,36,28,43,35,50,42,57,49,64,56,71,63,78,70,85,77,92,
%T A168211 84,99,91,106,98,113,105,120,112,127,119,134,126,141,133,148,140,155,
%U A168211 147,162,154,169,161,176,168,183,175,190,182,197,189,204,196,211,203
%N A168211 a(n)=7*n-a(n-1)+1 (with a(1)=0)
%e A168211 For n=2, a(2)=7*2-0+1=15; n=3, a(3)=7*3-15+1=7; n=4, a(4)=7*4-7+1=22
%K A168211 nonn,new
%O A168211 1,2
%A A168211 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168235
%S A168235 13,39,79,133,201,283,379,489
%N A168235 It is well known that there are fields in which the fundamental theorem of arithmetic breaks down. One such is k(sqrt(-5)). Hence 6 can be factorised in two different ways: a)2*3 and b) (1+sqrt(-5))*(1-sqrt(-5)). In the context of the property of polynomials refered to in A165806, A165808, A165809 and related sequences the present sequence is the sequence of quotients when f(x) is the cyclotomic quadratic polynomial x^2 + x + 1 and the quotients are generated by f(x + n*f(x))/f(x) when x =2. In the next few sequences I will submit the quotients when x=3, (1 + sqrt(-5)) and (1 + sqrt(-5)) respectively. Note: n belongs to N.
%e A168235 When x =2, f(x) = 7. Hence f( x + f(x))/f()x) = 13 when n= 1.
%p A168235 In pari {p(n) = (2 + n*7)/7}; for (n = 1,10, print p(n))).
%Y A168235 Cf. A165806, A165808 and A165809.
%K A168235 nonn,uned,new
%O A168235 13,1
%A A168235 A. K. Devaraj (dkandadai(AT)gmail.com), Nov 21 2009

%I A168210
%S A168210 0,13,6,19,12,25,18,31,24,37,30,43,36,49,42,55,48,61,54,67,60,73,66,79,
%T A168210 72,85,78,91,84,97,90,103,96,109,102,115,108,121,114,127,120,133,126,
%U A168210 139,132,145,138,151,144,157,150,163,156,169,162,175,168,181,174,187
%N A168210 a(n)=6*n-a(n-1)+1 (with a(1)=0)
%e A168210 For n=2, a(2)=6*2-0+1=13; n=3, a(3)=6*3-13+1=6; n=4, a(4)=6*4-6+1=19
%K A168210 nonn,new
%O A168210 1,2
%A A168210 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168214
%S A168214 1,11,51,192,669,2222,7135,22374,68916,209348,628916,1872269,5531641,
%T A168214 16238866,47410139,137758585,398617683
%N A168214 Least k such that H(k) >= n, where H(k) is the harmonic number sum_{i=n..k} 1/i.
%H A168214 Northwolves, <a href="http://blog.csdn.net/northwolves/archive/2009/11/20/4844754.aspx">Harmonic number sum </a>
%e A168214 For n = 2 the a(2) = 11 means 1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11>=2
%o A168214 (Other) VBA Program: Sub Harmonic_number_sum() Dim s As Double, i As Long, j As Long, n As Long For n = 1 To 15 s = 0 For i = n To 1000000000 s = s + 1 / i If s >= n Then Exit For Next Debug.Print "a(" & n & ")=" & i: Next End Sub
%Y A168214 Cf. A004080, A002387
%K A168214 easy,nice,nonn,new
%O A168214 1,2
%A A168214 Zhining Yang (northwolves(AT)163.com), Nov 20 2009

%I A168206
%S A168206 0,11,5,16,10,21,15,26,20,31,25,36,30,41,35,46,40,51,45,56,50,61,55,66,
%T A168206 60,71,65,76,70,81,75,86,80,91,85,96,90,101,95,106,100,111,105,116,110,
%U A168206 121,115,126,120,131,125,136,130,141,135,146,140,151,145,156,150,161
%N A168206 a(n)=5*n-a(n-1)+1 (with a(1)=0)
%e A168206 For n=2, a(2)=5*2-0+1=11; n=3, a(3)=5*3-11+1=5; n=4, a(4)=5*4-5+1=16
%K A168206 nonn,new
%O A168206 1,2
%A A168206 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168203
%S A168203 0,9,4,13,8,17,12,21,16,25,20,29,24,33,28,37,32,41,36,45,40,49,44,53,48,
%T A168203 57,52,61,56,65,60,69,64,73,68,77,72,81,76,85,80,89,84,93,88,97,92,101,
%U A168203 96,105,100,109,104,113,108,117,112,121,116,125,120,129,124,133,128,137
%N A168203 a(n)=4*n-a(n-1)+1 (with a(1)=0)
%e A168203 For n=2, a(2)=4*2-0+1=9; n=3, a(3)=4*3-9+1=4; n=4, a(4)=4*4-4+1=13
%K A168203 nonn,new
%O A168203 1,2
%A A168203 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168204
%S A168204 1,8,5,12,9,16,13,20,17,24,21,28,25,32,29,36,33,40,37,44,41,48,45,52,49,
%T A168204 56,53,60,57,64,61,68,65,72,69,76,73,80,77,84,81,88,85,92,89,96,93,100,
%U A168204 97,104,101,108,105,112,109,116,113,120,117,124,121,128,125,132,129,136
%N A168204 a(n)=4*n-a(n-1)+1 (with a(1)=1)
%e A168204 For n=2, a(2)=4*2-1+1=8; n=3, a(3)=4*3-8+1=5; n=4, a(4)=4*4-5+1=12
%K A168204 nonn,new
%O A168204 1,2
%A A168204 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168197
%S A168197 0,7,3,10,6,13,9,16,12,19,15,22,18,25,21,28,24,31,27,34,30,37,33,40,36,
%T A168197 43,39,46,42,49,45,52,48,55,51,58,54,61,57,64,60,67,63,70,66,73,69,76,
%U A168197 72,79,75,82,78,85,81,88,84,91,87,94,90,97,93,100,96,103,99,106,102
%N A168197 a(n)=3*n-a(n-1)+1 (with a(1)=0)
%e A168197 For n=2, a)2)=3*2-0+1=7; n=3, a(3)=3*3-7+1=3; n=4, a(4)=3*4-3+1=10
%K A168197 nonn,new
%O A168197 1,2
%A A168197 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168239
%S A168239 6,5,4,12,11,3,2,6,7,7,10,5,0,1,7,3,9,8,2,6,12,5,13,8,11,3,12,8,12,5,14,
%T A168239 3,1,10,14,3,3,8,4,13,9,13,1,13,11,3,2,4,9,8,7,12,14,11,5,3,3,8,5,4,11,
%U A168239 13,13,2,14,3,12,8,15,3,6,3,4,12,6,8,4,8,14,13,2,7,9,7,2,7,13,8,4,3,9,8
%N A168239 Iterate the map n -> sum of largest digit of all divisors of n; sequence gives number of steps to reach 14.
%D A168239 J. Earls, "Black Hole 14," Mathematical Bliss, Pleroma Publications, 2009, pages 18-22. ASIN: B002ACVZ6O
%K A168239 base,easy,nonn,new
%O A168239 2,1
%A A168239 Jason Earls (zevi_35711(AT)yahoo.com), Nov 21 2009

%I A168198
%S A168198 1,6,4,9,7,12,10,15,13,18,16,21,19,24,22,27,25,30,28,33,31,36,34,39,37,
%T A168198 42,40,45,43,48,46,51,49,54,52,57,55,60,58,63,61,66,64,69,67,72,70,75,
%U A168198 73,78,76,81,79,84,82,87,85,90,88,93,91,96,94,99,97,102,100,105,103,108
%N A168198 a(n)=3*n-a(n-1)+1 (with a(1)=1)
%e A168198 For n=2, a(2)=3*2-1+1=6; n=3, a(3)=3*3-6+1=4; n=4, a(4)=3*4-4+1=9
%K A168198 nonn,new
%O A168198 1,2
%A A168198 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168218
%S A168218 6,0,5,5,2,1,7,8,8,8,8,2,6,0,0,4,4,7,6,9,9,5,4,9,0,0,5,2,0,7,2,4,0,4,4,
%T A168218 7,3,0,3,2,3,8,8,9,8,4,5,5,0,6,5,7,8,3,3,1,1
%N A168218 Decimal expansion of the sum_{k=2..infinity} 1/(k^2*log(k)).
%H A168218 R. P. Boas, Jr, <a href="http://www.jstor.org/stable/2318865">Partial sums of Infinite Series, and How They Grow</a>, Am. Math. Monthly 84 (4) (1977) 237-235 [<a href="http://www.ams.org/mathscinet-getitem?mr=440240">MR0440240</a>].
%e A168218 equals 1/(4*log(2))+1/(9*log(3))+1/(16*log(4))+ .... + = 0.605521788882600447699549005207240447303238898..
%K A168218 cons,nonn,new
%O A168218 0,1
%A A168218 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 20 2009

%I A168213
%S A168213 5,14,14,23,23,32,32,41,41,50,50,59,59,68,68,77,77,86,86,95,95,104,104,
%T A168213 113,113,122,122,131,131,140,140,149,149,158,158,167,167,176,176,185,
%U A168213 185,194,194,203,203,212,212,221,221,230,230,239,239,248,248,257,257
%N A168213 a(n)=9*n-a(n-1)+1 (with a(1)=5)
%e A168213 For n=2, a(2)=9*2-5+1=14; n=3, a(3)=9*3-14+1=14; n=4, a(4)=9*4-14+1=23
%K A168213 nonn,new
%O A168213 1,1
%A A168213 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168243
%S A168243 1,1,1,5,11,59,439,2659,13705,160649,2009681,16966421,183312931,
%T A168243 2078169235,34203787591,657685416179,8054585463569,104530824746129,
%U A168243 2595754682459425,39767021562661669,758079429084897211
%N A168243 Expansion of prod( (1+x^i)^(1/i), i = 1..infinity).
%F A168243 G.f.: exp(sum( A048272(n)*x^n/n, n = 1..infinity)).
%Y A168243 Cf. A028342.
%K A168243 easy,nonn,new
%O A168243 0,4
%A A168243 Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 21 2009

%I A168228
%S A168228 1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,5,10,10,5,1,1,5,4,25,4,5,1,
%T A168228 1,15,64,50,50,64,15,1,1,15,65,66,30,66,65,15,1,1,55,455,671,1410,
%U A168228 1410,671,455,55,1,1,55,1815,4730,11495,7251,11495,4730,1815,55
%V A168228 1,1,-1,1,-1,1,1,1,-1,-1,1,1,0,1,1,1,-5,10,-10,5,-1,1,-5,-4,25,-4,-5,1,
%W A168228 1,15,64,50,-50,-64,-15,-1,1,15,65,66,30,66,65,15,1,1,-55,455,-671,1410,
%X A168228 -1410,671,-455,55,-1,1,-55,1815,-4730,11495,-7251,11495,-4730,1815,-55
%N A168228 Coefficient triangle sequence of characteristic polynomials of a Fermat like matrix:M(n)=Pascal nth matrix: F(n)=Inverse[Transpose[M(n)]].M(n)
%C A168228 Row sums are:
%C A168228 {1, 0, 1, 0, 4, 0, 9, 0, 324, 0, 9801, 0,...}
%C A168228 Example Matrix F(3):
%C A168228 {{1, 1, 1},
%C A168228 {-1, -3, -2},
%C A168228 {1, 2, 1}}
%D A168228 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 172.
%e A168228 {1},
%e A168228 {1, -1},
%e A168228 {1, -1, 1},
%e A168228 {1, 1, -1, -1},
%e A168228 {1, 1, 0, 1, 1},
%e A168228 {1, -5, 10, -10, 5, -1},
%e A168228 {1, -5, -4, 25, -4, -5, 1},
%e A168228 {1, 15, 64, 50, -50, -64, -15, -1},
%e A168228 {1, 15, 65, 66, 30, 66, 65, 15, 1},
%e A168228 {1, -55, 455, -671, 1410, -1410, 671, -455, 55, -1},
%e A168228 {1, -55, 1815, -4730, 11495, -7251, 11495, -4730, 1815, -55, 1},
%e A168228 {1, 197, 4675, -33825, -54978, 99174, -99174, 54978, 33825, -4675, -197, -1}
%t A168228 Clear[T, M, F];
%t A168228 T[n_, m_] := If[n >= m, Binomial[n, m], 0];
%t A168228 M[n_] := Table[T[k, m], {k, 0, n}, {m, 0, n}];
%t A168228 F[n_] := Inverse[Transpose[M[n]]].M[n];
%t A168228 Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[F[n], x], x], {n, 0, 10}]];
%t A168228 Flatten[%]
%Y A168228 A045912
%K A168228 sign,uned,new
%O A168228 0,17
%A A168228 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 20 2009

%I A168245
%S A168245 1,5,7,17,19,33,33,45,63,69,95,105,109,125,147,171,165,209,219,225,255,
%T A168245 273,295,331,353,361,381,385,399,483,485,511,523,581,579,617,653,665,
%U A168245 697,717,729,791,789,815,813,899,987,987,993,1013,1033,1045,1115,1119
%N A168245 prime(prime(n+1))-2*prime(n).
%F A168245 a(n)=A006450(n+1)-A100484(n).
%K A168245 nonn,new
%O A168245 1,2
%A A168245 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009

%I A168224
%S A168224 0,5,7,11,17,19,23,29,31,35,41,43,47,53,55,59,65,67,71,77,79,83,89,91,
%T A168224 95,101,103,107,113,115,119,125,127,131,137,139,143,149,151,155,161,163,
%U A168224 167,173,175,179,185,187,191,197,199,203,209,211,215,221,223,227,233
%N A168224 Where record values occur in A168223.
%C A168224 A168223(a(n))=A047342(n) and A168223(m)<A047342(n) for m<a(n).
%K A168224 nonn,new
%O A168224 1,2
%A A168224 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009

%I A168202
%S A168202 5,2,8,5,11,8,14,11,17,14,20,17,23,20,26,23,29,26,32,29,35,32,38,35,41,
%T A168202 38,44,41,47,44,50,47,53,50,56,53,59,56,62,59,65,62,68,65,71,68,74,71,
%U A168202 77,74,80,77,83,80,86,83,89,86,92,89,95,92,98,95,101,98,104,101,107,104
%N A168202 a(n)=3*n-a(n-1)+1 (with a(1)=5)
%e A168202 For n=2, a(2)=3*2-5+1=2; n=3, a(3)=3*3-2+1=8; n=4, a(4)=3*4-8+1=5
%K A168202 nonn,new
%O A168202 1,1
%A A168202 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168195
%S A168195 5,0,7,2,9,4,11,6,13,8,15,10,17,12,19,14,21,16,23,18,25,20,27,22,29,24,
%T A168195 31,26,33,28,35,30,37,32,39,34,41,36,43,38,45,40,47,42,49,44,51,46,53,
%U A168195 48,55,50,57,52,59,54,61,56,63,58,65,60,67,62,69,64,71,66,73
%N A168195 a(n)=2*n-a(n-1)+1 (with a(1)=5)
%F A168195 a(n)=2*n-a(n-1)+1 (with a(1)=5)
%e A168195 For n=2, a(2)=2*2-5+1=0; n=3, a(3)=2*3-0+1=7; n=4, a(4)=2*4-7+1=2
%K A168195 nonn,new
%O A168195 1,1
%A A168195 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168238
%S A168238 1,4,29,263,2724
%N A168238 Number of different 0-moment rowing configurations for 4n rowers.
%C A168238 Also the number of ways to assign 2n +1's and 2n -1's to the numbers 1,2,...,4n such that the sum is 0, assuming 1 gets the sign +1.
%D A168238 John D. Barrow, "Rowing and the Same-Sum Problem Have Their Moments", preprint, http://arxiv.org/pdf/0911.3551v1
%e A168238 For n = 2 there are 4-solutions to the 8-man rowing problem.
%K A168238 nonn,new
%O A168238 1,2
%A A168238 Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Nov 21 2009

%I A168212
%S A168212 4,11,11,18,18,25,25,32,32,39,39,46,46,53,53,60,60,67,67,74,74,81,81,88,
%T A168212 88,95,95,102,102,109,109,116,116,123,123,130,130,137,137,144,144,151,
%U A168212 151,158,158,165,165,172,172,179,179,186,186,193,193,200,200,207,207
%N A168212 a(n)=7*n-a(n-1)+1 (with a(1)=4)
%e A168212 For n=2, a(2)=7*2-4+1=11; n=3, a(3)=7*3-11+1=11; n=4, a(4)=7*4-11+1=18
%K A168212 nonn,new
%O A168212 1,1
%A A168212 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168251
%S A168251 0,1,4,9,64,25,576,49,4096,81,25600,121,147456,169,802816,225
%N A168251 A168077=0,1,1,9,4,25,9,. Take array of mix A168077(2n)*2^n , A168077(2n+1) successive values. a(n) is main diagonal. Square.
%C A168251 Row 1) 0,1,1,9,4,25,=A168077; 2) 0,1,2,9,8,25,=A129194; 3) 0,1,4,9,16,25,=A000290; 4) 0,1,8,9,32,25,; 5) 0,1,16,9,64,25,=A154615. Via A129194,a(n) is linked to Rydberg-Ritz spectra of hydrogen.
%F A168251 Mix A128782 , A016754 .
%K A168251 nonn,uned,new
%O A168251 0,3
%A A168251 Paul Curtz (bpcrtz(AT)free.fr), Nov 21 2009

%I A168015
%S A168015 1,1,4,6,13,12,29,22,46,48,72,67,137
%N A168015 a(n) = A000041(n) + n*A032741(n).
%Y A168015 Cf. A000005, A032741, A000041, A168014, A168020, A168021.
%K A168015 easy,more,nonn,new
%O A168015 0,3
%A A168015 Omar E. Pol (info(AT)polprimos.com), Nov 20 2009

%I A167524
%S A167524 1,4,6,7,8,9,11,12,13,15,17,19,30,40,42,46,48,49,60,62,64,66,67
%N A167524 Lexicographically smallest sequence which lists the position of the odd digits in the concatenation of its terms.
%e A167524 We have a(1)=1 <=> the sequence starts with an odd digit.
%e A167524 a(2) can't be 2 (since this would place an even digit in position 2 and at the same time state that there's an odd digit), nor can it be 3 (since then the second odd digit would be in position 2 and not in position a(2)). But a(2)=4 is possible.
%e A167524 This implies that there follows another even digit, a(3)=6, before the next odd digit, a(4)=7, etc.
%e A167524 ______________________ 1 _________________ 2 _________________ 3 _________________ 4
%e A167524 pos. 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 ...
%e A167524 seq. 1,4,6,7,8,9,1 1,1 2,1 3,1 5,1 7,1 9,3 0,4 0,4 2,4 6,4 8,4 9,6 0,6 2,6 4,6 6,6 7,...
%K A167524 base,more,nonn,new
%O A167524 1,2
%A A167524 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 21 2009

%I A168233
%S A168233 1,4,4,7,7,10,10,13,13,16,16,19,19,22,22,25,25,28,28,31,31,34,34,37,37,
%T A168233 40,40,43,43,46,46,49,49,52,52,55,55,58,58,61,61,64,64,67,67,70,70,73,
%U A168233 73,76,76,79,79,82,82,85,85,88,88,91,91,94,94,97,97,100,100,103,103,106
%N A168233 a(n)=3*n-a(n-1)-1 (with a(1)=1)
%e A168233 For n=2, a(2)=3*2-1-1=4; n=3, a(3)=3*3-4-1=4; n=4, a(4)=3*4-4-1=7
%K A168233 nonn,new
%O A168233 1,2
%A A168233 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009

%I A168200
%S A168200 4,3,7,6,10,9,13,12,16,15,19,18,22,21,25,24,28,27,31,30,34,33,37,36,40,
%T A168200 39,43,42,46,45,49,48,52,51,55,54,58,57,61,60,64,63,67,66,70,69,73,72,
%U A168200 76,75,79,78,82,81,85,84,88,87,91,90,94,93,97,96,100,99,103,102,106,105
%N A168200 a(n)=3*n-a(n-1)+1 (with a(1)=4)
%e A168200 For n=2, a(2)=3*2-4+1=3; n=3, a(3)=3*3-3+1=7; n=4, a(4)=3*4-7+1=6
%K A168200 nonn,new
%O A168200 1,1
%A A168200 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168230
%S A168230 0,4,1,5,2,6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14,11,15,12,16,13,17,14,
%T A168230 18,15,19,16,20,17,21,18,22,19,23,20,24,21,25,22,26,23,27,24,28,25,29,
%U A168230 26,30,27,31,28,32,29,33,30,34,31,35,32,36,33,37,34,38,35,39,36,40,37
%N A168230 a(n) = n+2-a(n-1) for n > 1; a(1) = 0.
%C A168230 Interleaving of A001477 and A000027 without first three terms.
%C A168230 a(n+1)-a(n) = A168309(n).
%C A168230 Binomial transform of 0, 4 followed by a signed version of A005009.
%C A168230 Inverse binomial transform of A034007 without first and third term.
%F A168230 a(n) = a(n-2)+1 for n > 2; a(1) = 0, a(2) = 4.
%F A168230 a(n) = (7*(-1)^n+2*n+5)/4.
%F A168230 G.f.: x^2*(4-3*x)/((1+x)*(1-x)^2).
%e A168230 a(2) = 2+2-a(1) = 4-0 = 4; a(3) = 3+2-a(2) = 5-4 = 1.
%o A168230 (MAGMA) [ n eq 1 select 0 else -Self(n-1)+n+2: n in [1..75] ];
%Y A168230 Cf. A001477 (nonnegative integers), A000027 (positive integers), A168309 (repeat 4,-3), A005009 (7*2^n), A034007 (first differences of A045891).
%K A168230 nonn,new
%O A168230 1,2
%A A168230 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009
%E A168230 Edited, four comments, three formulae, Magma program added by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 22 2009

%I A168155
%S A168155 0,3,8,14,32,61,117,230,470,922,1807,3597,7071,14022,27693,54876,109077,
%T A168155 216301,430183,854696,1700412,3382868,6733230,13404811
%N A168155 Sum of binary digits of all primes < 2^n, i.e., with at most n binary digits.
%C A168155 Partial sums of A168156.
%F A168155 a(n) = A095375( pi( 2^n-1 )), where pi = A000720.
%e A168155 No prime can be written with only 1 binary digit, thus a(1)=0.
%e A168155 The primes that can be written with 2 binary digits are 2 = 10[2] and 3 = 11[2], they have 3 nonzero bits, so a(2)=3.
%e A168155 Primes with 3 binary digits are 5 = 101[2] and 7 = 111[3]. They add 5 more nonzero bits to yield a(3) = a(2)+5 = 8.
%o A168155 (PARI) s=0; L=p=2; while( L*=2, print1(s", "); until( L<p=nextprime(p+1), s+=norml2(binary(p))))
%Y A168155 Cf. A168153.
%K A168155 more,nonn,new
%O A168155 1,2
%A A168155 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 20 2009

%I A168209
%S A168209 3,8,8,13,13,18,18,23,23,28,28,33,33,38,38,43,43,48,48,53,53,58,58,63,
%T A168209 63,68,68,73,73,78,78,83,83,88,88,93,93,98,98,103,103,108,108,113,113,
%U A168209 118,118,123,123,128,128,133,133,138,138,143,143,148,148,153,153,158
%N A168209 a(n)=5*n-a(n-1)+1 (with a(1)=3)
%e A168209 For n=2, a(2)=5*2-3+1=8; n=3, a(3)=5*3-8+1=8; n=4, a(4)=5*4-8+1=13
%K A168209 nonn,new
%O A168209 1,1
%A A168209 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168241
%S A168241 1,3,6,10,13,15,19,21,24,27,29,32,35,38,41,43,47,51,54,56,59,61,63,66,
%T A168241 69,71,74,76,78,82,85,90,93,96,100,102,104,107,110,114,116,118,121,124,
%U A168241 126,129,131,135,137,140,144,146,149,151,153,157,160,163,165,168,171
%N A168241 n-th square-free number plus n-th non-single or nonisolated number.
%F A168241 a(n)=A005117(n)+A167707(n).
%K A168241 nonn,new
%O A168241 1,2
%A A168241 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009

%I A168252
%S A168252 1,3,5,7,10,11,13,14,15,17,19,21,22,26,29,31,33,34,35,38,39,41,43,46,51,
%T A168252 55,57,58,59,61,62,65,66,69,70,71,73,74,77,78,82,85,86,87,91,93,94,95,
%U A168252 101,103,105,106,107,109,110,111
%N A168252 Numbers n such that n=square-free and n=non-single or nonisolated.
%Y A168252 Cf. A005117(square-free numbers), A167707(non-single or nonisolated numbers).
%K A168252 nonn,new
%O A168252 1,2
%A A168252 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009

%I A168156
%S A168156 0,3,5,6,18,29,56,113,240,452,885,1790,3474,6951,13671,27183,54201,
%T A168156 107224,213882,424513,845716,1682456,3350362,6671581,13299828
%N A168156 Sum of the binary digits of all primes between 2^(n-1) and 2^n-1, i.e., with exactly n binary digits.
%C A168156 Sequence A168155 yields the partial sums.
%e A168156 No prime can be written with only 1 binary digit, thus a(1)=0.
%e A168156 The primes that can be written with 2 binary digits are 2 = 10[2] and 3 = 11[2], they have 3 nonzero bits, so a(2)=3.
%e A168156 Primes with 3 binary digits are 5 = 101[2] and 7 = 111[3]. They have a total of a(3)=5 nonzero bits.
%o A168156 (PARI) s=0; L=p=2; while( L*=2, print1(s", "); s=0; until( L<p=nextprime(p+1), s+=norml2(binary(p))))
%K A168156 more,nonn,new
%O A168156 1,2
%A A168156 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 20 2009

%I A167525
%S A167525 3,5,6,7,8,20,21,31,33,35,37,39,40,51,53,55,57,58,70,72,74,76,81
%N A167525 Lexicographically smallest sequence which lists the position of the even digits in the concatenation of its terms.
%C A167525 "Lexicographically smallest" refers to comparing sequences term by term, not the strings obtained by the concatenation. (This is not possible, since then the 1st term could be an arbitrarily long string of 1's.) In other words, term after term, the smallest possible value not leading to a contradiction is appended.
%e A167525 We can't have a(1)=1 (since then the 1st digit would not be even) nor a(1)=2 (since then the 1st digit would be even), but a(1)=3 is possible.
%e A167525 This implies that there follows another odd digit, a(2)=5, before the first even digit a(3)=6.
%e A167525 Then comes another odd digit a(4)=7, since the second even digit occurs only in position a(2)=5, namely a(5)=8.
%e A167525 ______________________ 1 _________________ 2 _________________ 3 _________________ 4
%e A167525 pos. 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 ...
%e A167525 seq. 3,5,6,7,8,2 0,2 1,3 1,3 3,3 5,3 7,3 9,4 0,5 1,5 3,5 5,5 7,5 8,7 0,7 2,7 4,7 6,8 1,...
%K A167525 base,more,nonn,new
%O A167525 1,1
%A A167525 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 21 2009

%I A168223
%S A168223 0,0,0,0,1,3,5,4,1,1,2,7,10,7,2,1,3,10,15,11,3,2,4,14,20,
%T A168223 14,4,2,5,17,25,18,5,3,6,21,30,21,6,3,7,24,35,25,7,4,8,
%U A168223 28,40,28,8,4,9,31,45,32,9,5,10,35,50,35,10,5,11,38,55,39
%V A168223 0,0,0,0,-1,3,-5,4,-1,-1,-2,7,-10,7,-2,-1,-3,10,-15,11,-3,-2,-4,14,-20,
%W A168223 14,-4,-2,-5,17,-25,18,-5,-3,-6,21,-30,21,-6,-3,-7,24,-35,25,-7,-4,-8,
%X A168223 28,-40,28,-8,-4,-9,31,-45,32,-9,-5,-10,35,-50,35,-10,-5,-11,38,-55,39
%N A168223 A006369(n) - A006368(n).
%C A168223 A047342 and A168223 give record values and where they occur: a(A168224(n))=A047342(n) and a(m)<A047342(n) for m<A168224(n).
%H A168223 R. Zumkeller, <a href="b168223.txt">Table of n, a(n) for n = 0..10000</a>
%F A168223 a(12*n)=-10*n, a(12*n+1)=7*n, a(12*n+2)=-2*n, a(12*n+3)=-n, a(12*n+4)=-2*n-1, a(12*n+5)=7*n+3, a(12*n+6)=-10*n-5, a(12*n+7)=7*n+4, a(12*n+8)=-2*n-1, a(12*n+9)=-n-1, a(12*n+10)=-2*n-2, a(12*n+11)=7*n+7.
%K A168223 nonn,new
%O A168223 0,6
%A A168223 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009

%I A168219
%S A168219 1,3,4,6,15,16,18,24,27,30,31,36,37,43,51,52,57,60,73,75,81,82,87,90,93,
%T A168219 106,108,109,114,145,154,159,160,163,165,171,174,175,178,196,201,204,
%U A168219 207,208,211,220,222,225,228,234
%N A168219 Naturals n for which 1 + 10*n^3 (A168147) is prime
%C A168219 (1) It is conjectured that sequence is infinite
%C A168219 (2) It is an open problem if 3 consecutive naturals n exist which give such a prime
%D A168219 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A168219 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D A168219 Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996
%e A168219 (1) 1+10*1^3=11 gives a(1)=1
%e A168219 (2) 1+10*3^3=271=3^4 gives a(2)=3
%e A168219 (3) 1+10*37^3=506531 gives a(13)=37
%Y A168219 Cf. A000040 The prime numbers
%Y A168219 Cf. A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
%Y A168219 Cf. A167535 Concatenation of two square numbers which give a prime
%K A168219 nonn,new
%O A168219 1,2
%A A168219 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 20 2009

%I A168234
%S A168234 3,3,13,13,13,13,13,13,13,13,39,39,39,39,39,39,39,39,39,39,39,39,39,39,
%T A168234 39,39,39,39,89,89,89,89,89,89,89,89,89,89,89,89,89,89,89,89,89,89,89,
%U A168234 89,89,89,89,89,89,89,89,89,89,89,89,89,171,171,171,171,171,171,171
%N A168234 From Janet form: (A138100=1,2,5,6,7,) + (2,1,8,7,6=A168142) . Extended.
%C A168234 First of two parts (odd and even rows). Note 3,13,39,89,171=(b(n)=2,12,38,88,170) + (1=A000004);b(n) is first bisection of future 2,4,12,20,38,56,88,120,170, extended Janet table first column (see A134984,A138725,submitted A168208).
%K A168234 nonn,uned,new
%O A168234 1,1
%A A168234 Paul Curtz (bpcrtz(AT)free.fr), Nov 21 2009

%I A168237
%S A168237 0,3,3,6,6,9,9,12,12,15,15,18,18,21,21,24,24,27,27,30,30,33,33,36,36,39,
%T A168237 39,42,42,45,45,48,48,51,51,54,54,57,57,60,60,63,63,66,66,69,69,72,72,
%U A168237 75,75,78,78,81,81,84,84,87,87,90,90,93,93,96,96,99,99,102,102,105,105
%N A168237 a(n)=3*n-a(n-1)-3 (with a(1)=0)
%e A168237 For n=2, a(2)=3*2-0-3=3; n=3, a(3)=3*3-3-3=3; n=4, a(4)=3*4-3-3=6
%K A168237 nonn,new
%O A168237 1,2
%A A168237 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009

%I A168242
%S A168242 1,1,1,1,3,1,1,29,21,1,1,1769,1872,265,1,1,615871,1227274,
%T A168242 405530,5975,1,1,1124878187,14488841796,8196098025,250869024,
%U A168242 194883,1,1,10146337387971,4127467195814962,2684293284374908
%V A168242 1,1,-1,1,-3,1,1,-29,21,-1,1,-1769,1872,-265,1,1,-615871,1227274,
%W A168242 -405530,5975,-1,1,-1124878187,14488841796,-8196098025,250869024,
%X A168242 -194883,1,1,-10146337387971,4127467195814962,-2684293284374908
%N A168242 Coefficient triangle sequence of characteristic polynomials of a Fermat like matrix:M(n)=Eulerian nth matrix: F(n)=M(n).Transpose[M(n)]]
%C A168242 Row Sums are:
%C A168242 {1, 0, -1, -8, -160, 211848,5418539727, 1568832578459224, 5586114023994799591396,
%C A168242 -416657044755533539390036118560, -14128568453639379750002082917435886133265, -6849588404991830798534151932598866046674668061916984...}.
%C A168242 Example matrix F(3);
%C A168242 {{3, 5, 1},
%C A168242 {5, 17, 4},
%C A168242 {1, 4, 1}}
%D A168242 L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 172
%e A168242 {1},
%e A168242 {1, -1},
%e A168242 {1, -3, 1},
%e A168242 {1, -29, 21, -1},
%e A168242 {1, -1769, 1872, -265, 1},
%e A168242 {1, -615871, 1227274, -405530, 5975, -1},
%e A168242 {1, -1124878187, 14488841796, -8196098025, 250869024, -194883, 1},
%e A168242 {1, -10146337387971, 4127467195814962, -2684293284374908, 136124480159052, -319484649614, 8897703, -1},
%e A168242 {1, -428247824076704947, 17914574195515702665641, -13896536495233638792866, 1577809812703562226910, -9306019885601872094, 778719385854493, -533785743, 1},
%e A168242 {1, -81122352002183205692001, 1013200645627301010738899491371, -2171326301305554075331374889621, 748623927505727977255573077308, -7156540983035789703544116336, 1305525584958241674283545, -3205618448985143519, 40926870693, -1},
%e A168242 {1, -66885549293275814462767924797, 700501791981375381769184102256200146631, -28023249349569184787979186261051800365390, 13350702011670792795258543143392523163311, -156586293448931879782269330892154553133, 63386232099122631948920609947972715, -438644853880761289559728814794, 21273733616165855220783, -3893960978593, 1},
%e A168242 {1, -234732980513118576078558469331193281, 5621271025686372941421637129418789177249005788203, -13305553109795291932551487280535074672188567064406267, 6564379514567042192853283682636347534279562461093950, -114178043014292921637849211194413456481495109216258, 141964308173916470071468540296775755854414852854, -2083148469963100139599682004755602288051786, 278086488638019757478602360279335075, -214476940412956125417947983, 450564987828509, -1}
%t A168242 Clear[t, T, M, F];
%t A168242 t[n_, k_] = Sum[(-1)^j Binomial[n, j](k + 1 - j)^(n - 1), {j, 0, k}];
%t A168242 T[n_, k_] := If[n >= m, t[n + 2, k], 0];
%t A168242 M[n_] := Table[T[m, k], {k, 0, n}, {m, 0, n}];
%t A168242 F[n_] := M[n].Transpose[M[n]];
%t A168242 Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[F[n], x], x], {n, 0, 10}]];
%t A168242 Flatten[%]
%K A168242 sign,uned,new
%O A168242 0,5
%A A168242 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2009

%I A168216
%S A168216 1,1,1,1,3,1,1,8,5,1,1,23,19,7,1,1,74,69,34,9,1,1,262,256,147,53,11,1,1,
%T A168216 993,986,615,265,76,13,1,1,3943,3935,2571,1235,431,103,15,1,1,16178,
%U A168216 16169,10862,5591,2216,653,134,17,1
%N A168216 Riordan array (1/(1-x),xc(x)/(1-xc(x))) where c(x)is the g.f. of A000108.It factorizes as A007318*A106566.
%C A168216 Inverse is Riordan array (1/(1+x-x^2),x(1-x)/(1+x-x^2)) = [ -1,-1,1,0,0,0,0,0,...]DELTA[1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A091698.
%F A168216 Sum_{k, 0<=k<=n}T(n,k)*x^k = A000012(n), A007317(n+1), A026375(n) for x = 0, 1, 2 respectively.
%e A168216 Triangle begins : 1 ; 1,1 ; 1,3,1 ; 1,8,5,1 ; 1,23,19,7,1 ; ...
%Y A168216 Cf. A000108, A007318, A106566.
%K A168216 nonn,tabl,new
%O A168216 0,5
%A A168216 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 20 2009

%I A166852
%S A166852 2,1036,2770
%N A166852 Numbers n such that n^n+3 is prime.
%C A166852 Numbers corresponding to a(2) and a(3) are probable primes. 2770 is in the
%C A166852 sequence so 2770^2770+3 is probable prime, it is interesting that 277027703
%C A166852 is also prime. For the first term we have the same property. Namely both
%C A166852 numbers 2^2+3 and 223 are prime.
%C A166852 Prime corresponding to the next term, if it exists has more than 20000 digits.
%t A166852 Do[If[GCD[n,3]==1&&PrimeQ[n^n+3],Print[n]],{n,2,5362,2}]
%Y A166852 Cf. A100407, A100408, A087037, A166853.
%K A166852 hard,more,nonn,new
%O A166852 1,1
%A A166852 Farideh Firoozbakht (mymontain(AT)yahoo.com), Nov 20 2009

%I A168207
%S A168207 2,14,7,19,12,24,17,29,22,34,27,39,32,44,37,49,42,54,47,59,52,64,57,69,
%T A168207 62,74,67,79,72,84,77,89,82,94,87,99,92,104,97,109,102,114,107,119,112,
%U A168207 124,117,129,122,134,127,139,132,144,137,149,142,154,147,159,152,164
%N A168207 a(n)=5*n-a(n-1)+1 (with a(1)=2)
%e A168207 For n=2, a(2)=5*2-2+1=9; n=3, a(3)=5*3-9+1=7; n=4, a(4)=5*4-7+1=14
%K A168207 nonn,new
%O A168207 1,1
%A A168207 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168205
%S A168205 2,7,6,11,10,15,14,19,18,23,22,27,26,31,30,35,34,39,38,43,42,47,46,51,
%T A168205 50,55,54,59,58,63,62,67,66,71,70,75,74,79,78,83,82,87,86,91,90,95,94,
%U A168205 99,98,103,102,107,106,111,110,115,114,119,118,123,122,127,126,131,130
%N A168205 a(n)=4*n-a(n-1)+1 (with a(1)=2)
%e A168205 For n=2, a(2)=4*2-2+1=7; n=3, a(3)=4*3-7+1=6; n=4, a(4)4*4-6+1=11
%K A168205 nonn,new
%O A168205 1,1
%A A168205 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168194
%S A168194 0,2,6,12,22,38,64,106,174,284,462,750,1216,1970,3190,5164,8358,13526,
%T A168194 21888
%N A168194 Numbers n such that adding 4 to two successive terms produce the next term.
%F A168194 a(n)= a(n-2)+ a(n-1)+ 4; n>2; a(1)= 0; a(2)= 2.
%e A168194 Example: n=3, a(3)= a(3-2)+ a(3-1)+ 4 = a(1)+ a(2)+ 4 = 0 + 2 + 4 = 6; n=4, a(4)= a(4-2)+a(4-1)+ 4 = a(2) + a(3)+4; = 2 + 6 + 4 = 12.
%K A168194 nonn,new
%O A168194 1,2
%A A168194 Geoffrey O. Ahiakwo (obuusoltd(AT)yahoo.com), Nov 19 2009

%I A168247
%S A168247 1,2,6,7,12,46,48,49,71,126,147,162,172,189,211,223,257,272,306,309,337
%N A168247 Numbers n of the form (prime(prime(n))-prime(n))/10.
%Y A168247 Cf. A000027, A168152.
%K A168247 nonn,new
%O A168247 1,2
%A A168247 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009

%I A168199
%S A168199 2,5,5,8,8,11,11,14,14,17,17,20,20,23,23,26,26,29,29,32,32,35,35,38,38,
%T A168199 41,41,44,44,47,47,50,50,53,53,56,56,59,59,62,62,65,65,68,68,71,71,74,
%U A168199 74,77,77,80,80,83,83,86,86,89,89,92,92,95,95,98,98,102,102,105,105,108
%N A168199 a(n)=3*n-a(n-1)+1 (with a(1)=2)
%e A168199 For n=2, a(2)=3*2-2+1=5; n=3, a(3)=3*3-5+1=5; n=4, a(4)=3*4-5+1=8
%K A168199 nonn,new
%O A168199 1,1
%A A168199 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 20 2009

%I A168246
%S A168246 1,2,4,19,92,576,4156,34178,314368,3199936,35703996,433422071,
%T A168246 5687955724,80256879068,1211781887796,19496946568898,333041104402860,
%U A168246 6019770247224496,114794574818830716,2303332661419442569
%N A168246 Inverse Weigh transform of n!.
%F A168246 Prod_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} n! x^n.
%Y A168246 Cf. A112354.
%K A168246 easy,nonn,new
%O A168246 1,2
%A A168246 Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 21 2009

%I A168154
%S A168154 2,4,5,8,10,18,28,29,33,54,57,58,61,64,66,76,86,88,93,114,128,133,141,
%T A168154 142,170,173,177,189,193,196,198,201,202,205,220,226,241,244,248,249,
%U A168154 253,265,269,276,277,280,292,297,326,330,352,376,377,389,400,409,416
%N A168154 Numbers n such that the sum of binary digits in prime(1),...,prime(n) is prime.
%C A168154 Indices of primes in A095375.
%o A168154 (PARI) s=0; for(n=1,999, isprime(s+=norml2(binary(prime(n)))) & print1(n", "))
%o A168154 is_A168154(n)=isprime(A095375(n))
%Y A168154 Cf. A168153.
%K A168154 nonn,new
%O A168154 1,1
%A A168154 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 20 2009

%I A168221
%S A168221 0,1,2,3,9,6,7,4,18,5,11,12,27,15,16,8,36,10,20,21,45,24,25,13,54,14,29,
%T A168221 30,63,33,34,17,72,19,38,39,81,42,43,22,90,23,47,48,99,51,52,26,108,28,
%U A168221 56,57,117,60,61,31,126,32,65,66,135,69,70,35,144,37,74,75,153,78,79,40
%N A168221 A006368(A006368(n)).
%C A168221 Inverse integer permutation to A168222;
%C A168221 a(A006369(n)) = A006368(n).
%H A168221 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A168221 nonn,new
%O A168221 0,3
%A A168221 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009

%I A168014
%S A168014 0,0,2,3,8,5,18,7,24,18,30,11,60
%N A168014 a(n) = A032741(n)*n.
%Y A168014 Cf. A000005, A032741, A000041, A168015, A168020, A168021.
%K A168014 easy,more,nonn,new
%O A168014 0,3
%A A168014 Omar E. Pol (info(AT)polprimos.com), Nov 20 2009

%I A168249
%S A168249 1,2,3,7,12,16,20,26,29,33,38,43,48,51,57,60,63,73,78,82,94,97,103,113,
%T A168249 119,124,126,131,137,146,151,160,170,173,176,182,192,196,201,205,211,
%U A168249 215,224,237,241,244,257,260,270,274,277,284,288,293,296,300,306,308
%N A168249 n-th single or isolated number minus n-th square-free number.
%F A168249 a(n)=A167706(n)-A005117(n).
%K A168249 nonn,new
%O A168249 1,2
%A A168249 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009

%I A168222
%S A168222 0,1,2,3,7,9,5,6,15,4,17,10,11,23,25,13,14,31,8,33,18,19,39,41,21,22,47,
%T A168222 12,49,26,27,55,57,29,30,63,16,65,34,35,71,73,37,38,79,20,81,42,43,87,
%U A168222 89,45,46,95,24,97,50,51,103,105,53,54,111,28,113,58,59,119,121,61,62
%N A168222 A006369(A006369(n)).
%C A168222 Inverse integer permutation to A168221;
%C A168222 a(A006368(n)) = A006369(n).
%H A168222 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A168222 nonn,new
%O A168222 0,3
%A A168222 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009

%I A168231
%S A168231 2,3,2,4,4,4,4,4,5,2,4,5,4,4,5,5,4,4,4,5,5,5,5,4,5,4,5,5,5,4,4,5,4,5,5,
%T A168231 5,5,5,4,5,5,5,5,4,5,5,5,5,5,4,4,5,5,5,5,5,4,5,5,4,4,5,5,5,5,7,5,4,5,5,
%U A168231 5,5,4,4,5,5,5,5,5,5,5,5,7,5,5,7,5,5,7,5,5,5,7,4,5,5,5,5,5,5,5,5,5,5,5
%N A168231 a(n)^2 = digital sum of (A076503(n))^2.
%C A168231 a(n)^2 = A007953((A076503(n))^2).
%Y A168231 Cf. A076503 Prime numbers whose squares have square digital sums, A061910 Numbers n such that sum of digits of n^2 is a square, A007953 Digital sum (i.e. sum of digits) of n, A000040 The prime numbers.
%K A168231 base,nonn,new
%O A168231 1,1
%A A168231 Zak Seidov (zakseidov(AT)yahoo.com), Nov 21 2009

%I A168236
%S A168236 2,2,5,5,8,8,11,11,14,14,17,17,20,20,23,23,26,26,29,29,32,32,35,35,38,
%T A168236 38,41,41,44,44,47,47,50,50,53,53,56,56,59,59,62,62,65,65,68,68,71,71,
%U A168236 74,74,77,77,80,80,83,83,86,86,89,89,92,92,95,95,98,98,101,101,104,104
%N A168236 a(n)=3*n-a(n-1)-2 (with a(1)=2)
%e A168236 For n=2, a(2)= 3*2-2-2=2; n=3, a(3)=3*3-2-2=5; n=4, a(4)=3*4-5-2=5
%K A168236 nonn,new
%O A168236 1,1
%A A168236 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009

%I A168208
%S A168208 1,2,2,1,2,2,2,3,2,4,2,5,2,6,2,7,2,8,2,8,1,2,8,2,2,8,3,2,8,4,2,8,5,2,8,
%T A168208 6,2,8,7,2,8,8,2,8,8,1,2,8,8,2,2,8,9,2,2,8,10,2,2,8,11,2,2,8,13,1,2,8,
%U A168208 13,2,2,8,14,2,2,8,15,2,2,8,16,2,2,8,18,1,2,8,18,2,2,8,18,3,2,8,18,4,2
%N A168208 Periodic elements A000027 written with electrons number by energy level.
%C A168208 From third,sum of consecutive terms gives natural numbers A000027.Element 120=2+8+18+32+32+18+8+2. Number of terms for every element: A093907=2,8,8,18,18,32,32,50, times A000027=1,2,3,4,5,6,7,8,;element 118=2+8+18+32+32+18+8 (last with 7 terms). Then for Mendeleyev-Moseley-Seaborg , but Janet .. .
%K A168208 nonn,uned,new
%O A168208 1,2
%A A168208 Paul Curtz (bpcrtz(AT)free.fr), Nov 20 2009

%I A168244
%S A168244 2,1,8,19,34
%V A168244 2,-1,-8,-19,-34
%N A168244 In continuation of the recent sequences (cf A168235 & A168240) this is the sequence of quotients generated by (x + n*f(x))/f(x) when f(x) = x^2 + x + 1 and x = (1 + sqrt(-5)). Note: n belongs to N.
%C A168244 The quotient consists of two parts: i) rational integer & ii) rational integer multiples of sqrt(-5).
%e A168244 (x + f(x))/f(x) ( when x = (1 + sqrt(-5)) = 2 + 5*sqrt(-5). This sequence is a sequence of only the rational integers of the quotients.
%o A168244 Computations were by hand.
%Y A168244 Cf. A168235, A168240, A165806, A165808 & A165809
%K A168244 nonn,uned,new
%O A168244 1,1
%A A168244 A. K. Devaraj (dkandadai(AT)gmail.com), Nov 21 2009

%I A168177
%S A168177 1,1,1,2,1,4,1,4,2,4,1,5,4,4,4,4,2,7,2,8,4,3,3,7,3,3,5,6,5,7,3,7,5,6,3,
%T A168177 10,5,8,4,8,7,7,9,6,5,5,4,11,5,6,3,6,4,9,6,11,5,2,4,15,2,6,5,8,5,7,4,5,
%U A168177 7,9,2,13,7,5,8
%N A168177 Number of prime factors of n!+2^n-1, counted with multiplicity.
%D A168177 Florian Luca and Igor E. Shparlinski, On the largest prime factor of n! + 2n - 1. Journal de Theorie des Nombres de Bordeaux 17 (2005), 859-870.
%H A168177 F. Luca and I. E. Shparlinski, <a href="http://www.emis.de/journals/JTNB/2005-3/article10.pdf">On the largest prime factor of n! + 2n - 1.</a>
%e A168177 6!+2^6-1 = 783 = 3^3*29, hence a(6) = 4.
%o A168177 (MAGMA) pfmult := func< n | &+[ d[2]: d in Factorization(n) ] >; [ pfmult(Factorial(n)+2^n-1): n in [1..50] ]; //Some values were computed using Dario Alpern's ECM Factorization Program.
%Y A168177 Cf. A127986 (n!+2^n-1), A139024 (number of distinct prime factors), A139023 (smallest prime factor), A127987 (largest prime factor).
%K A168177 more,nonn,new
%O A168177 1,4
%A A168177 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 20 2009

%I A168217
%S A168217 1,1,2,1,3,12,3,10,48,224,10,42,226,1620,9040,40,245,1530,10024,95904,
%T A168217 720192,245,1365,10892,93096,744528,8855616,87805824,1225,11326,87696,
%U A168217 799344,8702064,87478464,1179952128,14662445184,11326,80094,836556
%N A168217 A triangular sequence based on the first level sum of polynomial coefficients: p(x,n,m)=(1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, {k, 0, Infinity}]/4
%C A168217 Row sums are:
%C A168217 {1, 3, 16, 285, 10938, 827935, 97511566, 15939477431, 3455244975656, 959962443311656,...}
%C A168217 This set of polynomials is a pure Infinite sum analogy to the Beta[n,m] integral.
%C A168217 Absolute values are used since the sums are zero otherwise.
%F A168217 p(x,n,m)=(1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, {k, 0, Infinity}]/4
%e A168217 {1},
%e A168217 {1, 2},
%e A168217 {1, 3, 12},
%e A168217 {3, 10, 48, 224},
%e A168217 {10, 42, 226, 1620, 9040},
%e A168217 {40, 245, 1530, 10024, 95904, 720192},
%e A168217 {245, 1365, 10892, 93096, 744528, 8855616, 87805824},
%e A168217 {1225, 11326, 87696, 799344, 8702064, 87478464, 1179952128, 14662445184},
%e A168217 {11326, 80094, 836556, 8401344, 88416672, 1166821632, 14621202720, 214725774528, 3224633430784},
%e A168217 {73626, 855162, 7965636, 92284896, 1140890112, 14497754256, 213099779232, 3217766464832, 51230717283840, 905285119960064}
%t A168217 p[x_, n_, m_] = (1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, { k, 0, Infinity}]/4;
%t A168217 Flatten[Table[Table[Apply[Plus, Abs[CoefficientList[ FullSimplify[ExpandAll[p[x, n, m]]], x]]], {m, 1, n}], {n, 1, 10}]]
%K A168217 nonn,uned,new
%O A168217 1,3
%A A168217 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 20 2009

%I A168021
%S A168021 1,2,1,3,0,1,5,2,0,1,7,0,0,0,1,11,3,2,0,0,1,15,0,0,0,0,0,1,22,5,0,2,0,0,
%T A168021 0,1,30,0,3,0,0,0,0,0,1,42,7,0,0,2,0,0,0,0,1,56,0,0,0,0,0,0,0,0,0,1,77,
%U A168021 11,5,3,0,2,0,0,0,0,0,1
%N A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.
%C A168021 In general, it appears that the number of partitions of n is also the number of partitions of n*k into parts divisible by k, for k>0.
%C A168021 In the square array, note that the column k lists each partition number of positive integers followed by k-1 zeroes. See A000041, which is the main entry for this sequence. Also see A168020.
%H A168021 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D)</a>
%H A168021 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A168021 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A168021 A000041(n) = number of partitions of (n*k) into parts divisible by k, for k>0.
%e A168021 Triangle begins:
%e A168021 ==============================================
%e A168021 ...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
%e A168021 ==============================================
%e A168021 n=1 ..... 1,
%e A168021 n=2 ..... 2, 1,
%e A168021 n=3 ..... 3, 0, 1,
%e A168021 n=4 ..... 5, 2, 0, 1,
%e A168021 n=5 ..... 7, 0, 0, 0, 1,
%e A168021 n=6 .... 11, 3, 2, 0, 0, 1,
%e A168021 n=7 .... 15, 0, 0, 0, 0, 0, 1,
%e A168021 n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
%e A168021 n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
%e A168021 n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
%e A168021 n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A168021 n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
%e A168021 ...
%Y A168021 Cf. A000041, A035377, A035444, A135010, A138121, A168020.
%K A168021 easy,more,nonn,tabl,new
%O A168021 1,2
%A A168021 Omar E. Pol (info(AT)polprimos.com), Nov 20 2009, Nov 21 2009

%I A168201
%S A168201 1,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,
%T A168201 1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,
%U A168201 1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,2,1,1,2,1,1,1,2,1,1,2,2,1,1,2,1,1
%N A168201 Number of representations of n in the form 7*k+11*m (with non-negative k, m).
%Y A168201 Cf. A168134, A168135.
%K A168201 nonn,new
%O A168201 0,78
%A A168201 Zak Seidov (zakseidov(AT)yahoo.com), Nov 20 2009

%I A168016
%S A168016 1,1,2,1,0,3,1,0,2,5,1,0,0,0,7,1,0,0,2,3,11,1,0,0,0,0,0,15,1,0,0,0,2,0,
%T A168016 5,22,1,0,0,0,0,0,3,0,30,1,0,0,0,0,2,0,0,7,42,1,0,0,0,0,0,0,0,0,0,56,1,
%U A168016 0,0,0,0,0,2,0,3,5,11,77
%N A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k, in deacreasing order.
%C A168016 In general, it appears that the partition number p(n,k) is the number of partitions of n into parts divisible by k, for k>0. Note that, for k=1 we can write p(n,k) = p(n) = A000041(n).
%C A168016 Also, it appears that the partition number p(n) is also the number of partitions of n*k into parts divisible by k.
%e A168016 Triangle begins:
%e A168016 n=1 ..... 1,
%e A168016 n=2 ..... 1, 2,
%e A168016 n=3 ..... 1, 0, 3,
%e A168016 n=4 ..... 1, 0, 2, 5,
%e A168016 n=5 ..... 1, 0, 0, 0, 7,
%e A168016 n=6 ..... 1, 0, 0, 2, 3, 11,
%e A168016 n=7 ..... 1, 0, 0, 0, 0, 0, 15,
%e A168016 n=8 ..... 1, 0, 0, 0, 2, 0, 5, 22,
%e A168016 n=9 ..... 1, 0, 0, 0, 0, 0, 3, 0, 30,
%e A168016 n=10 .... 1, 0, 0, 0, 0, 2, 0, 0, 7, 42,
%e A168016 n=11 .... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
%e A168016 n=12 .... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
%e A168016 ..................................................
%e A168016 Or better:
%e A168016 ==============================================
%e A168016 .... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
%e A168016 ==============================================
%e A168016 n=1 ....................................... 1,
%e A168016 n=2 ................................... 1 , 2,
%e A168016 n=3 ............................... 1 , 0 , 3,
%e A168016 n=4 ............................ 1, 0 , 2 , 5,
%e A168016 n=5 ......................... 1, 0, 0 , 0 , 7,
%e A168016 n=6 ...................... 1, 0, 0, 2 , 3, 11,
%e A168016 n=7 ................... 1, 0, 0, 0, 0 , 0, 15,
%e A168016 n=8 ................ 1, 0, 0, 0, 2, 0 , 5, 22,
%e A168016 n=9 ............. 1, 0, 0, 0, 0, 0, 3 , 0, 30,
%e A168016 n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0 , 7, 42,
%e A168016 n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 56,
%e A168016 n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5 ,11, 77,
%e A168016 ...
%Y A168016 Cf. A000005, A000041, A035363, A035444, A135010, A138121, A168014, A168015, A168020, A168021.
%K A168016 easy,more,nonn,tabl,new
%O A168016 1,3
%A A168016 Omar E. Pol (info(AT)polprimos.com), Nov 21 2009

%I A168229
%S A168229 1,2,0,9,4,2,9,2,0,2,8,8,8,1,8,9
%N A168229 Digits of decimal expansion of arctan(sqrt(7)).
%C A168229 See this constant used on p.6 of Cvijovic. Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function Cl_2(theta) by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.
%H A168229 Djurdje Cvijovic, <a href="http://arxiv.org/abs/0911.3773">A dilogarithmic integral arising in quantum field theory</a>, Nov 29, 2009.
%e A168229 1.209429202888189.
%K A168229 cons,more,nonn,new
%O A168229 1,2
%A A168229 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 20 2009

%I A168232
%S A168232 2,0,4,2,6,4,8,6,10,8,12,10,14,12,16,14,18,16,20,18,22,20,24,22,26,24,
%T A168232 28,26,30,28,32,30,34,32,36,34,38,36,40,38,42,40,44,42,46,44,48,46,50,
%U A168232 48,52,50,54,52,56,54,58,56,60,58,62,60,64,62,66,64,68,66,70,68,72
%N A168232 a(n)=2*n-a(n-1)-2 (with a(1)=2)
%e A168232 For n=2, a(2)=2*2-2-2=0; n=3, a(3)=2*3-0-2=4; n=4, a(4)=2*4-4-2=2
%K A168232 nonn,new
%O A168232 1,1
%A A168232 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009

%I A168020
%S A168020 1,2,0,3,1,0,5,0,0,0,7,2,1,0,0,11,0,0,0,0,0,15,3,0,1,0,0,0,22,0,2,0,0,0,
%T A168020 0,0,30,5,0,0,1,0,0,0,0,42,0,0,0,0,0,0,0,0,0,56,7,3,2,0,1,0,0,0,0,0,77,
%U A168020 0,0,0,0,0,0,0,0,0,0,0,101,11,0,0,0,0,1,0,0,0,0,0,0,135,0,5,0,2,0,0,0,0
%N A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.
%C A168020 In general, it appears that the number of partitions of n is also the number of partitions of n*k into parts divisible by k, for k>0.
%C A168020 In the square array, note that the column k starts with k-1 zeroes. Then lists each partition number of positive integers followed by k-1 zeroes. See A000041, which is the main entry for this sequence.
%H A168020 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D)</a>
%H A168020 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A168020 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A168020 A000041(n) = number of partitions of (n*k) into parts divisible by k, for k>0.
%e A168020 The array begins:
%e A168020 ================================================================
%e A168020 ..... Column k: 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 .10 .11 .12
%e A168020 Row ..........................................................
%e A168020 .n ...........................................................
%e A168020 ===============================================================
%e A168020 ... 1 ....... ..1, .0, .0, .0, .0, .0, .0, .0, .0, .0, .0, .0,
%e A168020 ... 2 ....... ..2, .1, .0, .0, .0, .0, .0, .0, .0, .0, .0, .0,
%e A168020 ... 3 ....... ..3, .0, .1, .0, .0, .0, .0, .0, .0, .0, .0, .0,
%e A168020 ... 4 ....... ..5, .2, .0, .1, .0, .0, .0, .0, .0, .0, .0, .0,
%e A168020 ... 5 ....... ..7, .0, .0, .0, .1, .0, .0, .0, .0, .0, .0, .0,
%e A168020 ... 6 ....... .11, .3, .2, .0, .0, .1, .0, .0, .0, .0, .0, .0,
%e A168020 ... 7 ....... .15, .0, .0, .0, .0, .0, .1, .0, .0, .0, .0, .0,
%e A168020 ... 8 ....... .22, .5, .0, .2, .0, .0, .0, .1, .0, .0, .0, .0,
%e A168020 ... 9 ....... .30, .0, .3, .0, .0, .0, .0, .0, .1, .0, .0, .0,
%e A168020 .. 10 ....... .42, .7, .0, .0, .2, .0, .0, .0, .0, .1, .0, .0,
%e A168020 .. 11 ....... .56, .0, .0, .0, .0, .0, .0, .0, .0, .0, .1, .0,
%e A168020 .. 12 ....... .77, 11, .5, .3, .0, .2, .0, .0, .0, .0, .0, .1,
%e A168020 ...
%Y A168020 Cf. A000041, A035377, A035444, A135010, A138121.
%K A168020 easy,more,nonn,tabl,new
%O A168020 1,2
%A A168020 Omar E. Pol (info(AT)polprimos.com), Nov 20 2009
%E A168020 Edited by Omar E. Pol (info(AT)polprimos.com), Nov 21 2009

%I A168019
%S A168019 1,1,1,2,0,1,3,1,0,1,5,0,0,0,1,7,2,1,0,0,1,11,0,0,0,0,0,1,15,3,0,1,0,0,
%T A168019 0,1,22,0,2,0,0,0,0,0,1,30,5,0,0,1,0,0,0,0,1,42,0,0,0,0,0,0,0,0,0,1,56,
%U A168019 7,3,2,0,1,0,0,0,0,0,1
%N A168019 Square array T(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k, for k>0.
%C A168019 Note that column k lists each partition number A000041 followed by k-1 zeroes. See also A168020 and A168021.
%e A168019 The array begins:
%e A168019 ==================================================
%e A168019 ... Column k: 1. 2. 3. 4. 5. 6. 7. 8. 9 10 11 12
%e A168019 . Row ...........................................
%e A168019 ...n ............................................
%e A168019 ==================================================
%e A168019 .. 0 ........ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e A168019 .. 1 ........ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e A168019 .. 2 ........ 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e A168019 .. 3 ........ 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e A168019 .. 4 ........ 5, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
%e A168019 .. 5 ........ 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
%e A168019 .. 6 ....... 11, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0,
%e A168019 .. 7 ....... 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
%e A168019 .. 8 ....... 22, 5, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0,
%e A168019 .. 9 ....... 30, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0,
%e A168019 . 10 ....... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0,
%e A168019 . 11 ....... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
%e A168019 . 12 ....... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
%e A168019 ...
%Y A168019 Cf. A000005, A000041, A035363, A035444, A047968, A035377, A135010, A168016, A168120, A168121.
%K A168019 easy,more,nonn,tabl,new
%O A168019 0,4
%A A168019 Omar E. Pol (info(AT)polprimos.com), Nov 21 2009

%I A175034
%S A175034 2,5,7,11,12,14,17,18,20,23,27,29,31,32,37,38,40,41,42,44,47,50,51,52,56,
%T A175034 57,59,62,65,67,68,69,70,73,74,77,82,83,84,86,87,88,92,95,96,98,101,102,
%U A175034 104,107,109,110,112,113,117,119,122,125,126,127,128,131,132,135,137,139
%N A175034 Offsets i such that i + n*(n+1)/2 is never a perfect square for any n>0.
%C A175034 Complement of A175035.
%Y A175034 Cf. A001108, A006451, A154138, A154139, A154140, A154141
%K A175034 easy,nonn,new
%O A175034 1,1
%A A175034 Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 10 2009
%E A175034 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 26 2009

%I A175035
%S A175035 1,3,4,6,8,9,10,13,15,16,19,21,22,24,25,26,28,30,33,34,35,36,39,43,45,46,
%T A175035 48,49,53,54,55,58,60,61,63,64,66,71,72,75,76,78,79,80,81,85,89,90,91,93,
%U A175035 94,97,99,100,103,105,106,108,111,114,115,116,118,120
%N A175035 Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.
%C A175035 The ansatz n*(n+1)/2+i=s^2 can be transformed into (2*n+1)^2-2*(2*s)^2 =1-8*i.
%C A175035 A necessary condition for solutions to this diophantine equation is that
%C A175035 D=2 is a quadratic residue of the squarefree part of 8*i-1 (see A057126).
%C A175035 A sufficient condition is then available by a sequence of tests on the
%C A175035 continued fractions of a quadratic surd that originates from a solution of this congruence.
%C A175035 See Mollin and Matthews for details. [R. J. Mathar, Nov 16 2009]
%H A175035 R. A. Mollin, <a href="http://dx.doi.org/10.1016/S0723-0869(01)80015-3">Simple continued fraction solutions for Diophantine Equations</a>, Exposit. Mathem. 19 (2001) 55-73.
%H A175035 Keith Matthews, <a href="http://www.numbertheory.org/pdfs/patz5.pdf">The Diophantine equation x^2-Dy^2=N,D >0</a>, Exposit. Mathem. 18 (4) (2000) 323-331 [<a href="http://www.ams.org/mathscinet-getitem?mr=1788328">MR1788328</a>].
%Y A175035 Cf. A001108, A006451, A154138 to A154154.
%K A175035 easy,nonn,new
%O A175035 1,2
%A A175035 Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 10 2009
%E A175035 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 26 2009

%I A078778
%S A078778 4,5,6,7,8,10,13,14,19,20,24,25,26,28,34,38,48,54,55,59,71,75,92,109
%N A078778 Numbers n such that n!+1 is a semiprime.
%C A078778 Note that the two prime factors of 38!+1 = 523022617466601111760007224100074291200000001 = 14029308060317546154181 * 37280713718589679646221 both have 23 decimal digits. Are there any other terms in this sequence other than 4,5,7 and 38 with this property?
%C A078778 It is likely that 114 and 115 are the next terms. [Sean A. Irvine (sairvin(AT)xtra.co.nz), Nov 15 2009]
%e A078778 4 is in the sequence because 4!+1=25=5*5 is semiprime. But 9 is not in the sequence because 9!+1=19*71*269 is not semiprime. [Sean A. Irvine (sairvin(AT)xtra.co.nz), Nov 15 2009]
%o A078778 (PARI) { fp(a,b)=local(c,d,r); for(n=a,b,r=n!+1; c=vecmin(factor(r)[, 1]~); d=vecmax(factor(r)[,1]~); if(bigomega(r)==2 && isprime(c) && isprime(d), print1(n" ");)) } fp(1,100)
%Y A078778 Cf. A001358, A082952, A090159, A090160, A078781.
%K A078778 more,nonn,new
%O A078778 1,1
%A A078778 Jason Earls (zevi_35711(AT)yahoo.com), Jan 09 2003
%E A078778 One more term (109) from Sean A. Irvine (sairvin(AT)xtra.co.nz), Nov 15 2009

%I A113610
%S A113610 0,4,6,18,3,18,9,9,40,3,35,27,6,27,43,55,6,50,36,9,65,27,70,84,36,3,18,
%T A113610 9,9,93,27,40,12,81,6,50,53,36,58,70,9,126,12,45,18,68,83,27,12,18,55,6,
%U A113610 99,55,58,70,9,65,45,12,135,147,27,6,27,126,50,99,15,27,70,84,68,80,36
%N A113610 Sum of digits of all composite numbers between prime(n) and prime(n+1).
%C A113610 Conjecture: For n > 9 if a(n) > a(n+1) < a(n+2) then prime(n) and prime(n+1) form a twin prime pair.
%e A113610 2 '' 3 '4' 5 '6' 7 '8910' 11 '12' 13 '141516' 17 '18' 19 '202122' 23 '2425262728' 29,...
%t A113610 Table[Total[Flatten[IntegerDigits/@Range[Prime[n]+1,Prime[n+1]-1]]],{n, 200}] # Zak Seidov (zakseidov(AT)yahoo.com), Nov 13 2009
%Y A113610 Cf. A175038 [Zak Seidov (zakseidov(AT)yahoo.com), Nov 13 2009]
%K A113610 base,easy,nonn,new
%O A113610 1,2
%A A113610 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 09 2005
%E A113610 More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006

%I A161219
%S A161219 4,6,8,12,16,28,40,72,120,216,376,704,1264,2364,4384,8232,15424,29204,55192,
%T A161219 104976,199760,381492,729448,1398504,2684368,5162856,9942136,19175160,37025584,
%U A161219 71585136,138547336,268439592,520602352,1010588256,1963413664,3817763800
%N A161219 (1/n) * Sum_{d|n} phi(n/d)*2^(d+1).
%Y A161219 Cf. A000031, A053635, A160619, A161217.
%K A161219 nonn,new
%O A161219 1,1
%A A161219 N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A161217
%S A161217 0,4,12,32,56,128,192,400,640,1232,2304,4496,8608,16960,33456,66304,132096,
%T A161217 263168,526320,1049872,2100352,4196480,8393904,16779152,33565952,67111488,
%U A161217 134235840,268441424,536906720,1073744960,2147560704,4294970896,8590069760
%N A161217 Sum_{d|n} phi(n/d)^2*2^(d+1).
%Y A161217 Cf. A053635.
%K A161217 nonn,new
%O A161217 0,2
%A A161217 N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A160620
%S A160620 0,2,6,16,28,64,96,200,320,616,1152,2248,4304,8480,16728,33152,66048,131584,
%T A160620 263160,524936,1050176,2098240,4196952,8389576,16782976,33555744,67117920,
%U A160620 134220712,268453360,536872480,1073780352,2147485448,4295034880,8589944384
%N A160620 Sum_{d|n} phi(n/d)^2*2^d.
%Y A160620 Cf. A053635.
%K A160620 nonn,new
%O A160620 0,2
%A A160620 N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A160619
%S A160619 0,4,12,24,48,80,168,280,576,1080,2160,4136,8448,16432,33096,65760,131712,
%T A160619 262208,525672,1048648,2099520,4194960,8392824,16777304,33564096,67109200,
%U A160619 134234256,268437672,536904480,1073741936,2147554080,4294967416,8590066944
%N A160619 Sum_{d|n} phi(n/d)*2^(d+1).
%Y A160619 Cf. A053635, A161219.
%K A160619 nonn,new
%O A160619 0,2
%A A160619 N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2009

%I A167451
%S A167451 2,9,10,11,12,99,990,1000,1001,1002,1003,1004,1005,1006,1007,1008,1010,
%T A167451 1011,1012,1013,1014,1015,1016,1017,1018,1020,1021,1022,1900,2000,2001,
%U A167451 2002,2003,2004,2005,2006,2007,2008,2010,2011,2012,2013,2014,2015,2016
%N A167451 Smallest sequence which lists the position of digits "9" in the sequence.
%C A167451 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "9" in the string obtained by concatenating all these terms, written in base 10.
%e A167451 We cannot have a(1)=1 (since then there's no "9" in the 1st place), but a(1)=2 is possible.
%e A167451 This implies that a(2) must start with a digit "9", so a(2)=9 is the smallest possible choice.
%e A167451 This allows us to go on with a(3)=10, a(4)=11, a(5)=12, but then must be follow 4 digits "9" (the 9th through 12th digit of the sequence), so a(6)=99 and a(7)=990 are the smallest possible choices.
%e A167451 Then the reasoning continues in analogy with A167450-A167457.
%o A167451 (PARI) concat([ [2,9,10,11,12,99,990], vector((99-11-1)\4,i,1000-(i<=9)+i+(i>=19)), [1900] , select(x->x%10-9 & x\10%10-9,vector((990-99)\4,i,2000-1+i)) ])
%o A167451 /* The following code checks a sequence for consistency (i.e., the given digit occurs exactly at positions given by the terms), but it does not check the monotonicity neither the minimality.
%o A167451 In case of a contradiction, it returns [n,pos,d] where n is the index of the term, pos is the position in the concatenation, and d is the digit for which the contradiction occurred.
%o A167451 If d is different from the given digit, the term a(n) said that there should be that digit at position pos, but we found d instead.
%o A167451 If d equals the given digit, we found d at position pos, but the term a(n) said that the next d should occur elsewhere. */
%o A167451 check_self(a,d=9)={ my(t=Vecsmall(concat(concat([""],a))),c=0); d+=48;
%o A167451 for( i=1,#a, a[i]>#t & break; t[a[i]]==d | return([i,a[i],t[a[i]]-48]));
%o A167451 for( i=1,#t, t[i]==d & (a[c++ ]==i | return([c,i,d-48]))) /* no contradiction => empty result */}
%Y A167451 Cf. A098645, A167519, A167520, A167450 - A167457.
%K A167451 base,nonn,new
%O A167451 1,1
%A A167451 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167450
%S A167450 2,8,9,10,11,88,880,900,901,902,903,904,905,906,907,909,910,911,912,913,
%T A167450 914,915,916,917,919,920,921,922,923,924,925,926,8000,9000,9001,9002,
%U A167450 9003,9004,9005,9006,9007,9009,9010,9011,9012,9013,9014,9015,9016,9017
%N A167450 Smallest sequence which lists the position of digits "8" in the sequence.
%C A167450 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "8" in the string obtained by concatenating all these terms, written in base 10.
%e A167450 We cannot have a(1)=1 (since then there's no "8" in the 1st place), but a(1)=2 is possible.
%e A167450 This implies that a(2) must start with a digit "8", so a(2)=8 is the smallest possible choice.
%e A167450 This allows us to go on with a(3)=9, a(4)=10, a(5)=11, but then must be follow 4 digits "8" (the 8th through 11th digit of the sequence), so a(6)=88 and a(7)=880 are the smallest possible choices.
%e A167450 Then the reasoning continues in analogy with A167452-A167457.
%o A167450 (PARI) concat([ [2,8,9,10,11,88,880], vector((88-11-1)\3,i,900-(i<=8)+i+(i>=18)), [8000], select(x->x%10-8 & x\10%10-8,vector((880-88)\4,i,9000-1+i)) ])
%Y A167450 Cf. A098645, A167519, A167520, A167451 - A167457.
%K A167450 base,nonn,new
%O A167450 1,1
%A A167450 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167457
%S A167457 2,7,8,9,10,77,770,800,801,802,803,804,805,806,808,809,810,811,812,813,
%T A167457 814,815,816,818,819,820,821,822,827,828,829,830,831,832,833,834,835,
%U A167457 836,838,839,840,841,842,843,844,845,846,848,849,850,851,852,853,854
%N A167457 Smallest sequence which lists the position of digits "7" in the sequence.
%C A167457 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "7" in the string obtained by concatenating all these terms, written in base 10.
%e A167457 We cannot have a(1)=1 (since then there's no "7" in the 1st place), but a(1)=2 is possible.
%e A167457 Then a(2) must start with a digit "7", so a(2)=7 is the smallest possible choice.
%e A167457 This allows us to go on with a(3)=8, a(4)=9, a(5)=10, but then must be follow 4 digits "6" (the 7th through 10th digit of the sequence), so a(6)=77 and a(7)=770 are the smallest possible choices.
%e A167457 Then the reasoning continues in analogy with A167452-A167456.
%o A167457 (PARI) concat([ [2,7,8,9,10,77,770], vector((77-10)\3-1,i,800-(i<=7)+i+(i>=17)), [827], select(x->x%10-7 & x\10%10-7,vector((770-77)\3+20,i,827+i)) ])
%Y A167457 Cf. A098645, A167519, A167520, A167452 - A167456.
%K A167457 base,nonn,new
%O A167457 1,1
%A A167457 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167456
%S A167456 2,6,7,8,9,66,660,700,701,702,703,704,705,707,708,709,710,711,712,713,
%T A167456 714,715,717,718,719,760,770,771,772,773,774,775,777,778,779,780,781,
%U A167456 782,783,784,785,787,788,789,790,791,792,793,794,795,797,798,799,800
%N A167456 Smallest sequence which lists the position of digits "6" in the sequence.
%C A167456 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "6" in the string obtained by concatenating all these terms, written in base 10.
%e A167456 We cannot have a(1)=1 (since then there's no "6" in the 1st place), but a(1)=2 is possible.
%e A167456 Then a(2) must start with a digit "6", so a(2)=6 is the smallest possible choice.
%e A167456 This allows us to go on with a(3)=7, a(4)=8, a(5)=9, but then must be follow 4 digits "6" (the 6th through 9th digit of the sequence), so a(6)=66 and a(7)=660 are the smallest possible choices.
%e A167456 Then the reasoning continues in analogy with A167452-A167455.
%o A167456 (PARI) concat([ [2,6,7,8,9,66,660], vector((66-9)\3-1,i,700-(i<=6)+i+(i>=16)), [760], select(x->x%10-6 & x\10%10-6,vector((660-66)\3+10,i,770+i-1)) ])
%Y A167456 Cf. A098645, A167519, A167520, A167452 - A167455.
%K A167456 base,nonn,new
%O A167456 1,1
%A A167456 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167455
%S A167455 2,5,6,7,55,56,60,61,62,63,64,66,67,68,69,70,71,72,73,74,76,77,78,79,80,
%T A167455 81,82,83,84,550,605,5555,6555,55555,56555,555555,600000,600001,600002,
%U A167455 600003,600004,600006,600007,600008,600009,600010,600011,600012,600013
%N A167455 Smallest sequence which lists the position of digits "5" in the sequence.
%C A167455 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "5" in the string obtained by concatenating all these terms, written in base 10.
%e A167455 We cannot have a(1)=1 (since then there's no "5" in the 1st place), but a(1)=2 is possible.
%e A167455 Then a(2) must start with a digit "5", so a(2)=5 is the smallest possible choice.
%e A167455 This allows us to go on with a(3)=6, a(4)=6, but then must be follow 3 digits "5" (the 5th, 6th and 7th digit of the sequence), so a(5)=55 and a(6)=56 are the smallest possible choice.
%e A167455 The reasoning continues in analogy with A167452-A167454.
%o A167455 (PARI) concat([ [2,5,6,7,55,56], vector((55-8)\2,i,60-(i<=5)+i+(i>=15)), [550, 605, 5555, 6555, 55 555, 56 555, 555 555], select(x->x%10-5 & x\10%10-5,vector((550-84)\6+10,i,600 000+i-1)) ])
%Y A167455 Cf. A098645, A167519, A167520, A167452, A167453, A167454.
%K A167455 base,nonn,new
%O A167455 1,1
%A A167455 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167454
%S A167454 2,4,5,44,50,51,52,53,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,400,
%T A167454 500,4444,5444,44444,45444,444000,500000,500001,500002,500003,500005,
%U A167454 500006,500007,500008,500009,500010,500011,500012,500013,500015,500016
%N A167454 Smallest sequence which lists the position of digits "4" in the sequence.
%C A167454 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "4" in the string obtained by concatenating all these terms, written in base 10.
%e A167454 We cannot have a(1)=1 (since then there's no "4" in the 1st place), but a(1)=2 is possible.
%e A167454 Then a(2)=4 is the smallest possible choice.
%e A167454 This allows us to take a(3)=5, but this must be followed by two digits "4" (the 4th and 5th of the sequence), thus a(4)=44. Terms a(5) through a(5+(44-6)/2) are now to be filled with 50,51,..., omitting terms with a digit "4".
%e A167454 The last term of this series is 70, which must be followed by 400 (whose 1st digit is the 44-th digit of the sequence), 500, and then 4444 (digits 50-53), 5444 (digits 54-57), 44444 (digits 58-62), 45444 (digits 63-67), 444000 (digits 68-73). This "predicts" that a(3) starts with a digit "3", so a(3)=30 is the smallest possible choice.
%e A167454 The next digit "3" must not appear until the 30th digit of the sequence, so we fill in terms 40,41,42,44,45... (omitting 43 which has a digit "3").
%e A167454 Now it happens that the term 53 would correspond to digits # 29 and 30=a(3) of the sequence, so we can simpliy continue with this and 4 more terms, up to 57.
%e A167454 The next term must have it's second digit (digit # 40=a(4) of the sequence) equal to 3, so 63 is the smallest choice.
%e A167454 The terms a(5)=41, a(6)=42 leave 330 as the smallest possible choice for the next term.
%e A167454 The terms 44,45,46 and 47,48,49,50 and 51,52,53,54,55 lead to the subsequent terms 333, 3333, 33333.
%o A167454 (PARI) concat([[2,4,5,44],vector((44-6)/2,i,50-(i<=4)+i+(i>=14)) ,[400,500,4444,5444,44 444,45 444, 444000], select(x->x%10-4 & x\10%10-4,vector((400-70)\6+10,i,500 000+i-1)) ])
%Y A167454 Cf. A098645, A167519, A167520, A167452, A167453.
%K A167454 base,nonn,new
%O A167454 1,1
%A A167454 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167453
%S A167453 2,3,30,40,41,42,44,45,46,47,48,49,50,51,52,53,54,55,56,57,63,330,333,
%T A167453 3333,33333,33400,40300,40400,40401,40402,40404,40405,40406,40407,40408,
%U A167453 40409,40410,40411,40412,40414,40415,40416,40417,40418,40419,40420
%N A167453 Smallest sequence which lists the position of digits "3" in the sequence.
%C A167453 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "3" in the string obtained by concatenating all these terms, written in base 10.
%e A167453 We cannot have a(1)=1 (since then there's no "3" in the 1st place), but a(1)=2 is possible.
%e A167453 Then a(2)=3 is a possible choice and certainly the smallest.
%e A167453 This "predicts" that a(3) starts with a digit "3", so a(3)=30 is the smallest possible choice.
%e A167453 The next digit "3" must not appear until the 30th digit of the sequence, so we fill in terms 40,41,42,44,45... (omitting 43 which has a digit "3").
%e A167453 Now it happens that the term 53 would correspond to digits # 29 and 30=a(3) of the sequence, so we can simpliy continue with this and 4 more terms, up to 57.
%e A167453 The next term must have it's second digit (digit # 40=a(4) of the sequence) equal to 3, so 63 is the smallest choice.
%e A167453 The terms a(5)=41, a(6)=42 leave 330 as the smallest possible choice for the next term.
%e A167453 The terms 44,45,46 and 47,48,49,50 and 51,52,53,54,55 lead to the subsequent terms 333, 3333, 33333.
%o A167453 (PARI) concat([[2,3,30],vector((40-4)/2-1,i,40-(i<=3)+i), [63, 330, 333, 3333, 33333, 33400,40300], select(x->x%10-3 & x\10%10-3,vector((330-63)\5+10,i,40400+i-1)) ])
%Y A167453 Cf. A098645, A167519, A167520, A167452.
%K A167453 base,nonn,new
%O A167453 1,1
%A A167453 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A168193
%S A168193 0,2,6,12,22,38,64,106,174,284,462,750,1216,1970,3190,5164,8358,13526,
%T A168193 21888
%N A168193 Numbers n such that adding 4 to two successive numbers results in the next number in succession.
%C A168193 a(1)= 0; a(2)= 2
%F A168193 a(1)= 0; a(2)= 2; a(n)= a(n-2)+ a(n-1)+4, n>2
%e A168193 Examples:1. n=3, a(3)= a(3-2)+ a(3-1)+ 4 = a(1)+ a(2)+ 4 = 0+2+4 = 6
%e A168193 . 2. n=4, a(4)= a(4-2)+ a(4-1)+ 4 = a(2)+ a(3)+ 4 = 2+6+4 = 12
%e A168193 .
%K A168193 nonn,new
%O A168193 1,2
%A A168193 Geoffrey O. Ahiakwo (obuusoltd(AT)yahoo.com), Nov 19 2009

%I A167452
%S A167452 3,4,22,30,31,33,34,35,36,37,38,42,43,44,45,52,202,222,223,302,2220,
%T A167452 3000,3200,3300,3301,3303,3304,3305,3306,3307,3308,3309,3310,3311,3313,
%U A167452 3314,3315,3316,3317,3318,3319,3330,3331,3333,3334,3335,3336,3337,3338
%N A167452 Smallest sequence which lists the position of digits "2" in the sequence.
%C A167452 The lexicographically smallest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "1" in the string obtained by concatenating all these terms, written in base 10.
%e A167452 We cannot have a(1)=1 (since then there's no 2 in the 1st place), nor a(1)=2 (since then the first occurrence of a "2" would be at position 1).
%e A167452 But a(1)=3 is possible, "predicting" that the first occurrence of a digit "2" will be the in the 3rd digit.
%e A167452 Then a(2)=4 is the smallest possible choice for a(2).
%e A167452 The next two digits (= the 3rd and 4th digit of the sequence) must be a "2", in view of a(1) and a(2). Thus a(3)=22 is the smallest possible choice.
%e A167452 This means that the next digit "2" will occur as the 22nd digit of the sequence, so the following terms are the least possible numbers without digit "2": 30,31,33,...,38. These make up digits 5 to 20 of the sequence.
%e A167452 The following number must have a "2" as second digit, the smallest possibility is 42.
%o A167452 (PARI) concat([ [3,4,22], vector((22-4)/2-1,i,i+30-(i<=2)), vector(4,i,42+i-1) , [52,202,222,223,302,2220,3000,3200], select(x -> x%10-2 & x\10%10-2 & x\100%10-2, vector((202-52)\4+13,i,3300+i-1)) ])
%Y A167452 Cf. A098645, A167519, A167520.
%K A167452 base,nonn,nice,new
%O A167452 1,1
%A A167452 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A168100
%S A168100 10,11,12,13,14,15,16,17,18,19,20,21,21,21,21,21,21,21,21,21,21,21,21,
%T A168100 21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21
%N A168100 a(n) = number of natural numbers m such that n - 10 <= m <= n + 10.
%C A168100 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 10 + n for 0 <= n <= 10, a(n) = 21 for n >= 11.
%K A168100 nonn,new
%O A168100 0,1
%A A168100 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168099
%S A168099 9,10,11,12,13,14,15,16,17,18,19,19,19,19,19,19,19,19,19,19,19,19,19,19,
%T A168099 19,19,19,19,19,19,19,19,19,19,19,19,19
%N A168099 a(n) = number of natural numbers m such that n - 9 <= m <= n + 9.
%C A168099 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 9 + n for 0 <= n <= 9, a(n) = 19 for n >= 10.
%K A168099 nonn,new
%O A168099 0,1
%A A168099 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168098
%S A168098 8,9,10,11,12,13,14,15,16,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,
%T A168098 17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17
%N A168098 a(n) = number of natural numbers m such that n - 8 <= m <= n + 8.
%C A168098 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 8 + n for 0 <= n <= 8, a(n) = 17 for n >= 9.
%K A168098 nonn,new
%O A168098 0,1
%A A168098 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168097
%S A168097 7,8,9,10,11,12,13,14,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,
%T A168097 15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15
%N A168097 a(n) = number of natural numbers m such that n - 7 <= m <= n + 7.
%C A168097 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 7 + n for 0 <= n <= 7, a(n) = 15 for n >= 8.
%K A168097 nonn,new
%O A168097 0,1
%A A168097 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168096
%S A168096 6,7,8,9,10,11,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,
%T A168096 13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13
%N A168096 a(n) = number of natural numbers m such that n - 6 <= m <= n + 6.
%C A168096 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 6 + n for 0 <= n <= 6, a(n) = 13 for n >= 7.
%K A168096 nonn,new
%O A168096 0,1
%A A168096 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168095
%S A168095 5,6,7,8,9,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,
%T A168095 11,11,11,11,11,11,11,11,11,11,11,11,11
%N A168095 a(n) = number of natural numbers m such that n - 5 <= m <= n + 5.
%C A168095 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 5 + n for 0 <= n <= 5, a(n) = 11 for n >= 6.
%K A168095 nonn,new
%O A168095 0,1
%A A168095 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168094
%S A168094 4,5,6,7,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%T A168094 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A168094 a(n) = number of natural numbers m such that n - 4 <= m <= n + 4.
%C A168094 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 4 + n for 0 <= n <= 4, a(n) = 9 for n >= 4.
%K A168094 nonn,new
%O A168094 0,1
%A A168094 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168093
%S A168093 3,4,5,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%T A168093 7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A168093 a(n) = number of natural numbers m such that n - 3 <= m <= n + 3.
%C A168093 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = 3 + n for 0 <= n <= 3, a(n) = 7 for n >= 4.
%K A168093 nonn,new
%O A168093 0,1
%A A168093 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168092
%S A168092 2,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%T A168092 5,5,5,5,5,5,5,5,5,5,5,5
%N A168092 a(n) = number of natural numbers m such that n - 2 <= m <= n + 2.
%C A168092 Generalisation: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see e.g. A158799). a(n) = (2 + n) for 0 <= n <= 2, a(n) = 5 for n >= 3.
%K A168092 nonn,new
%O A168092 0,1
%A A168092 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A161215
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%U A161215 40479533921803813920,327672500035999538160,602267704294826658336
%N A161215 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 17.
%D A161215 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161215 nonn,new
%O A161215 1,1
%A A161215 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161213
%S A161213 1,131071,64570081,8589869056,190734863281,8463265086751,38771752331201,
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%U A161213 554648540725313536,720867993281778161,5081852349802846271
%N A161213 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.
%D A161213 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161213 nonn,new
%O A161213 1,2
%A A161213 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161195
%S A161195 65535,2147385345,470177777355,35182761492480,499992370589085,
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%U A161195 2248845733577866995,16383250007092548195,27375595878265462275
%N A161195 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 16.
%D A161195 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161195 nonn,new
%O A161195 1,1
%A A161195 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161167
%S A161167 1,65535,21523360,2147450880,38146972656,1410533397600,5538821761600,
%T A161167 70367670435840,308836690967520,2499961853010960,4594972986357216,
%U A161167 46220358372556800,55451384098598320,362986684146456000
%N A161167 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.
%D A161167 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161167 nonn,new
%O A161167 1,2
%A A161167 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161157
%S A161157 32767,536821761,78361756228,4397643866112,49998474112902,
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%U A161157 819125001391673466,1244326279702202508,10516894943504990208
%N A161157 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 15.
%D A161157 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161157 nonn,new
%O A161157 1,1
%A A161157 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161139
%S A161139 1,32767,7174453,536854528,7629394531,235085301451,791260251657,
%T A161139 8795824586752,34315186290957,249992370597277,417724816941565,
%U A161139 3851637578973184,4265491084507563,25927224666044919,54736732481116543
%N A161139 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.
%D A161139 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161139 nonn,new
%O A161139 1,2
%A A161139 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161117
%S A161117 16383,134193153,13059888663,549655154688,4999694820123,106973548038633,
%T A161117 264555442913583,2251387513602048,6940560290953383,40952500271627493,
%U A161117 56558559305400519,438163652766240768,413500239275072043
%N A161117 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 14.
%D A161117 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161117 nonn,new
%O A161117 1,1
%A A161117 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161025
%S A161025 1,16383,2391484,134209536,1525878906,39179682372,113037178808,
%T A161025 1099444518912,3812797945332,24998474116998,37974983358324,
%U A161025 320959957991424,328114698808274,1851888100411464,3649114989636504
%N A161025 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.
%D A161025 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161025 nonn,new
%O A161025 1,2
%A A161025 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161024
%S A161024 8191,33542145,2176512520,68694312960,499938962796,8912818769400,
%T A161024 18895663909200,140685952942080,385562663380440,2047250052649620,
%U A161024 2570686683371352,18253452839731200,15902884603186140,77377743708174000
%N A161024 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 13.
%D A161024 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161024 nonn,new
%O A161024 1,1
%A A161024 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161010
%S A161010 1,8191,797161,33550336,305175781,6529545751,16148168401,137422176256,
%T A161010 423644039001,2499694822171,3452271214393,26745019396096,25239592216021,
%U A161010 132269647372591,243274230757741,562881233944576,619036127056621
%N A161010 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 13.
%D A161010 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161010 nonn,new
%O A161010 1,2
%A A161010 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A161004
%S A161004 4095,8382465,362706435,8583644160,49987791945,742460072445,
%T A161004 1349525501415,8789651619840,21417452280315,102325010111415,
%U A161004 116835129114795,760279114183680,611574734464785,2762478701396505
%N A161004 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 12.
%D A161004 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A161004 nonn,new
%O A161004 1,1
%A A161004 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160972
%S A160972 1,4095,265720,8386560,61035156,1088123400,2306881200,17175674880,
%T A160972 47071500840,249938963820,313842837672,2228476723200,1941507093540,
%U A160972 9446678514000,16218261652320,35175782154240,36413889826860
%N A160972 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 12.
%D A160972 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160972 nonn,new
%O A160972 1,2
%A A160972 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160964
%S A160964 2047,2094081,60435628,1072169472,4997558082,61825647444,96371138776,
%T A160964 548950769664,1189554465924,5112501917886,5309390815620,31654731491328,
%U A160964 23516361067738,98587674967848,147547904812968,281062794067968
%N A160964 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 11.
%D A160964 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160964 nonn,new
%O A160964 1,1
%A A160964 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160960
%S A160960 1,2047,88573,2096128,12207031,181308931,329554457,2146435072,
%T A160960 5230147077,24987792457,28531167061,185660345344,149346699503,
%U A160960 674597973479,1081213356763,2197949513728,2141993519227,10706111066619
%N A160960 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 11.
%D A160960 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160960 nonn,new
%O A160960 1,2
%A A160960 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160959
%S A160959 1023,522753,10067343,133824768,499511463,5144412273,6880289823,
%T A160959 34259140608,66051837423,255250357593,241218048687,1316969541888,
%U A160959 904033571463,3515828099553,4915692307383,8770339995648,7582212353463
%N A160959 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 10.
%D A160959 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160959 nonn,new
%O A160959 1,1
%A A160959 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160957
%S A160957 1,1023,29524,523776,2441406,30203052,47079208,268173312,581120892,
%T A160957 2497558338,2593742460,15463962624,11488207654,48162029784,72080070744,
%U A160957 137304735744,125999618778,594486672516,340614792100,1278749869056
%N A160957 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.
%D A160957 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160957 nonn,new
%O A160957 1,2
%A A160957 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160956
%S A160956 511,130305,1676080,16679040,49902216,427400400,490968800,2134917120,
%T A160956 3665586960,12725065080,10953738768,54707251200,34736533160,
%U A160956 125197044000,163679268480,273269391360,222788253240,934724674800
%N A160956 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 9.
%D A160956 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160956 nonn,new
%O A160956 1,1
%A A160956 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160953
%S A160953 1,511,9841,130816,488281,5028751,6725601,33488896,64566801,249511591,
%T A160953 235794769,1287360256,883708281,3436782111,4805173321,8573157376,
%U A160953 7411742281,32993635311,17927094321,63874967296,66186639441
%N A160953 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.
%D A160953 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160953 nonn,new
%O A160953 1,2
%A A160953 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160913
%S A160913 255,32385,278715,2072640,4980405,35396805,35000535,132648960,203183235,
%T A160913 632511435,496922835,2265395520,1333405965,4445067945,5443582665,
%U A160913 8489533440,6539772585,25804270845,12663182955,40480731840,38255584755
%N A160913 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 8.
%D A160913 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160913 nonn,new
%O A160913 1,1
%A A160913 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160908
%S A160908 1,255,3280,32640,97656,836400,960800,4177920,7173360,24902280,21435888,
%T A160908 107059200,67977560,245004000,320311680,534773760,435984840,1829206800,
%U A160908 943531280,3187491840,3151424000,5466151440,3559590240,13703577600
%N A160908 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.
%D A160908 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160908 nonn,new
%O A160908 1,2
%A A160908 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160898
%S A160898 127,8001,46228,256032,496062,2912364,2490216,8193024,11233404,31251906,
%T A160898 22498812,93195648,51083718,156883608,180566568,262176768,191591946,
%U A160898 707704452,331934820,1000060992,906438624,1417425156,854570808
%N A160898 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 7.
%D A160898 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160898 nonn,new
%O A160898 1,1
%A A160898 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160897
%S A160897 1,127,1093,8128,19531,138811,137257,520192,796797,2480437,1948717,
%T A160897 8883904,5229043,17431639,21347383,33292288,25646167,101193219,49659541,
%U A160897 158747968,150021901,247487059,154764793,568569856,305171875,664088461
%N A160897 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 7.
%D A160897 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160897 nonn,new
%O A160897 1,2
%A A160897 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160896
%S A160896 63,1953,7623,31248,49203,236313,176463,499968,617463,1525293,1014615,
%T A160896 3781008,1949283,5470353,5953563,7999488,5590683,19141353,8666343,
%U A160896 24404688,21352023,31453065,18431343,60496128,30751875,60427773
%N A160896 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 6.
%D A160896 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160896 nonn,new
%O A160896 1,1
%A A160896 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160895
%S A160895 1,63,364,2016,3906,22932,19608,64512,88452,246078,177156,733824,402234,
%T A160895 1235304,1421784,2064384,1508598,5572476,2613660,7874496,7137312,
%U A160895 11160828,6728904,23482368,12206250,25340742,21493836,39529728,21243690
%N A160895 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 6.
%D A160895 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160895 nonn,new
%O A160895 1,2
%A A160895 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160894
%S A160894 31,465,1240,3720,4836,18600,12400,29760,33480,72540,45384,148800,73780,
%T A160894 186000,193440,238080,161820,502200,224440,580320,496000,680760,394320,
%U A160894 1190400,604500,1106700,903960,1488000,783060,2901600,954304,1904640
%N A160894 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 5.
%D A160894 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160894 nonn,new
%O A160894 1,1
%A A160894 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160893
%S A160893 1,31,121,496,781,3751,2801,7936,9801,24211,16105,60016,30941,86831,
%T A160893 94501,126976,88741,303831,137561,387376,338921,499255,292561,960256,
%U A160893 488125,959171,793881,1389296,732541,2929531,954305,2031616,1948705
%N A160893 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.
%D A160893 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160893 nonn,new
%O A160893 1,2
%A A160893 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160892
%S A160892 15,105,195,420,465,1365,855,1680,1755,3255,1995,5460,2745,5985,6045,
%T A160892 6720,4605,12285,5715,13020,11115,13965,8295,21840,11625,19215,15795,
%U A160892 23940,13065,42315,14895,26880,25935,32235,26505,49140,21105,40005
%N A160892 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 4.
%D A160892 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160892 nonn,new
%O A160892 1,1
%A A160892 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160891
%S A160891 1,15,40,120,156,600,400,960,1080,2340,1464,4800,2380,6000,6240,7680,
%T A160891 5220,16200,7240,18720,16000,21960,12720,38400,19500,35700,29160,48000,
%U A160891 25260,93600,30784,61440,58560,78300,62400,129600,52060,108600,95200
%N A160891 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.
%D A160891 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160891 nonn,new
%O A160891 1,2
%A A160891 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160890
%S A160890 7,21,28,42,42,84,56,84,84,126,84,168,98,168,168,168,126,252,140,252,
%T A160890 224,252,168,336,210,294,252,336,210,504,224,336,336,378,336,504,266,
%U A160890 420,392,504,294,672,308,504,504,504,336,672,392,630,504,588,378,756
%N A160890 ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 3.
%D A160890 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160890 nonn,new
%O A160890 1,1
%A A160890 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160889
%S A160889 1,7,13,28,31,91,57,112,117,217,133,364,183,399,403,448,307,819,381,868,
%T A160889 741,931,553,1456,775,1281,1053,1596,871,2821,993,1792,1729,2149,1767,
%U A160889 3276,1407,2667,2379,3472,1723,5187,1893,3724,3627,3871,2257,5824,2793
%N A160889 Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 3.
%D A160889 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%K A160889 nonn,new
%O A160889 1,2
%A A160889 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160888
%S A160888 3,19,225,3475,61521
%N A160888 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160888 nonn,new
%O A160888 1,1
%A A160888 njas (njas(AT)research.att.com), Nov 15 2009

%I A160887
%S A160887 4,35,431,6267,102555
%N A160887 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160887 nonn,new
%O A160887 1,1
%A A160887 njas (njas(AT)research.att.com), Nov 15 2009

%I A160886
%S A160886 4,26,250,3086,44674
%N A160886 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160886 nonn,new
%O A160886 1,1
%A A160886 njas (njas(AT)research.att.com), Nov 15 2009

%I A160885
%S A160885 4,29,281,3231,42099
%N A160885 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160885 nonn,new
%O A160885 1,1
%A A160885 njas (njas(AT)research.att.com), Nov 15 2009

%I A160884
%S A160884 3,15,121,1271,15233
%N A160884 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160884 nonn,new
%O A160884 1,1
%A A160884 njas (njas(AT)research.att.com), Nov 15 2009

%I A160883
%S A160883 4,27,231,2251,23899
%N A160883 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160883 nonn,new
%O A160883 1,1
%A A160883 njas (njas(AT)research.att.com), Nov 15 2009

%I A160882
%S A160882 3,13,81,637,5649
%N A160882 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160882 nonn,new
%O A160882 1,1
%A A160882 njas (njas(AT)research.att.com), Nov 15 2009

%I A160881
%S A160881 3,14,85,591,4403
%N A160881 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160881 nonn,new
%O A160881 1,1
%A A160881 njas (njas(AT)research.att.com), Nov 15 2009

%I A160880
%S A160880 0,3,84,1995,45384
%N A160880 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160880 nonn,new
%O A160880 1,2
%A A160880 njas (njas(AT)research.att.com), Nov 15 2009

%I A160879
%S A160879 0,3,126,3255,72540
%N A160879 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160879 nonn,new
%O A160879 1,2
%A A160879 njas (njas(AT)research.att.com), Nov 15 2009

%I A160878
%S A160878 0,3,84,11755,33480
%N A160878 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160878 nonn,new
%O A160878 1,2
%A A160878 njas (njas(AT)research.att.com), Nov 15 2009

%I A160877
%S A160877 0,3,84,1680,29760
%N A160877 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160877 nonn,new
%O A160877 1,2
%A A160877 njas (njas(AT)research.att.com), Nov 15 2009

%I A160876
%S A160876 0,3,56,855,12400
%N A160876 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160876 nonn,new
%O A160876 1,2
%A A160876 njas (njas(AT)research.att.com), Nov 15 2009

%I A160875
%S A160875 0,3,84,1365,18600
%N A160875 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160875 nonn,new
%O A160875 1,2
%A A160875 njas (njas(AT)research.att.com), Nov 15 2009

%I A160874
%S A160874 0,3,42,465,4836
%N A160874 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160874 nonn,new
%O A160874 1,2
%A A160874 njas (njas(AT)research.att.com), Nov 15 2009

%I A160873
%S A160873 0,3,42,420,3720
%N A160873 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160873 nonn,new
%O A160873 1,2
%A A160873 njas (njas(AT)research.att.com), Nov 15 2009

%I A160872
%S A160872 0,3,28,195,1240
%N A160872 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.
%K A160872 nonn,new
%O A160872 1,2
%A A160872 njas (njas(AT)research.att.com), Nov 15 2009

%I A160871
%S A160871 1,63,7987,7987616,24875000437,193466859054994
%N A160871 Column 6 of array in A057004.
%D A160871 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%K A160871 nonn,new
%O A160871 1,2
%A A160871 njas (njas(AT)research.att.com), Nov 15 2009

%I A160870
%S A160870 1,1,1,1,3,1,1,4,7,1,1,7,13,15,1,1,6,35,40,31,1,1,12,31,155,121,63,
%T A160870 1,1,8,91,156,651,364,127,1,1,15,57,600,600,781,2667,1093,255,1
%N A160870 Array read by antidiagonals: T(n,k) = number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).
%D A160870 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%F A160870 T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
%e A160870 Array begins:
%e A160870 .1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e A160870 .1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
%e A160870 .1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
%e A160870 .1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
%e A160870 .1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
%e A160870 ....
%Y A160870 Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.
%Y A160870 Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.
%K A160870 nonn,tabl,easy,more,new
%O A160870 1,5
%A A160870 N. J. A. Sloane (njas(AT)research.att.com), Nov 19 2009

%I A160869
%S A160869 1,12,91,600,3751,22932,138811,836400,5028751,30203052
%N A160869 Row 6 of array in A160870.
%D A160869 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%K A160869 nonn,new
%O A160869 1,2
%A A160869 njas (njas(AT)research.att.com), Nov 15 2009

%I A160454
%S A160454 1,7,161,14721,1730861,207388305,24883501301
%N A160454 Number of isomorphism classes of multiple coverings of graphs with specified Betti number. See reference for precise definition.
%D A160454 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%K A160454 nonn,new
%O A160454 0,2
%A A160454 njas (njas(AT)research.att.com), Nov 15 2009

%I A160450
%S A160450 1,5,43,681,14491,336465,7997683
%N A160450 Number of isomorphism classes of multiple coverings of graphs with specified Betti number. See reference for precise definition.
%D A160450 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%K A160450 nonn,new
%O A160450 0,2
%A A160450 njas (njas(AT)research.att.com), Nov 15 2009

%I A167458
%S A167458 24,25,26,27,53,54,55,88,89,90,124,125,126,127,181,182,183,215,216,268,
%T A167458 269,270,271,303,304,305,337,338,339,340,341,342,343,344,345,346,347,
%U A167458 348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364
%N A167458 Indices of numbers in A167459 which are not in A066737.
%C A167458 Also, indices of terms in A167459 which are in A166505 (or: which are not in A152242).
%o A167458 (PARI) c=0; for(n=1,9999, is_A167459(n) & c++ & !is_A152242(n) & print1(c", "))
%Y A167458 Cf. A166504, A152242.
%K A167458 base,nonn,new
%O A167458 1,1
%A A167458 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167459
%S A167459 22,25,27,32,33,35,52,55,57,72,75,77,112,115,117,132,133,135,172,175,
%T A167459 177,192,195,202,203,205,207,213,217,219,222,225,231,232,235,237,243,
%U A167459 247,252,253,255,259,261,267,272,273,275,279,289,292,295,297,302,303
%N A167459 Composite numbers in A166504, i.e. whose decimal expression can be split up into prime numbers, with leading zeros allowed.
%C A167459 In contrast to A066737 (which is a subsequence of this one), we allow for leading zeros in the "prime" substrings; the two sequences differ from n=24 on, with a(24)=202 which is not in A066737.
%C A167459 Sequence A166505 gives the difference, A167459 \ A066737 = A166504 \ A152242. Sequence A167458 gives the indices of the terms not in A066737.
%F A167459 A167459 = A002808 n A166504, where "n" means intersection.
%F A167459 A167459 \ A066737 = A166504 \ A152242.
%o A167459 (PARI) is_A167459(n) = !isprime(n) & is_A166504(n)
%Y A167459 Cf. A002808, A066737, A121609, A166504, A167505.
%K A167459 base,nonn,new
%O A167459 1,1
%A A167459 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A168175
%S A168175 1,4,9,8,31,180,503,752,513,7316,25673,51480,26209,255524,1205559,
%T A168175 3033568,3695359,6453540,51681673,161551912,284435937,6880364,
%U A168175 1963530103,7902282960,17864421119,16141703756,60484132809
%V A168175 1,4,9,8,-31,-180,-503,-752,513,7316,25673,51480,26209,-255524,-1205559,
%W A168175 -3033568,-3695359,6453540,51681673,161551912,284435937,6880364,
%X A168175 -1963530103,-7902282960,-17864421119,-16141703756,60484132809
%N A168175 Binet's sequence:f(n) = (1/2 - I/Sqrt[3])*(2 + I*Sqrt[3])^n + (1/2 + I/Sqrt[3])*(2 - I*Sqrt[3])^n
%C A168175 The characteristic polynomial is 7 - 4 x + x^2.
%C A168175 Four different calculation methods are given in Mathematica.
%C A168175 From the section on "Discrete Servo Systems" in the reference
%C A168175 as related to transfer functions and Laplace transforms
%C A168175 used in control systems technology.
%D A168175 R. Pallu de la Barriere, Optimal Control Theory,Dover Publications, New York,1967,pages 266-7
%F A168175 f(n) = (1/2 - I/Sqrt[3])*(2 + I*Sqrt[3])^n + (1/2 + I/Sqrt[3])*(2 - I*Sqrt[3])^n
%t A168175 Clear[f, n, a, a0, q, x, t, v];
%t A168175 (*Binet*) f[n_] = (1/2 - I/Sqrt[3])*(2 + I*Sqrt[3])^n + (1/2 + I/Sqrt[3])*(2 - I*Sqrt[3])^n;
%t A168175 a0 = Table[FullSimplify[f[n]], {n, 0, 30}]
%t A168175 (* linear recursion *)
%t A168175 a[0] = 1; a[1] = 4;
%t A168175 a[n_] := a[n] = 4*a[n - 1] - 7*a[n - 2];
%t A168175 Table[a[n], {n, 0, 30}]
%t A168175 (* polynomial expansion with scale 7*)
%t A168175 q[x_] = 1/(x^2 - 4*x + 7);
%t A168175 Table[7^(n + 1)*SeriesCoefficient[ Series[q[t], {t, 0, 60}], n], {n, 0, 30}]
%t A168175 (* Matrix Markov recursion*)
%t A168175 v[0] = {1, 4};
%t A168175 m = {{0, 1}, {-7, 4}};
%t A168175 v[n_] := v[n] = m.v[n - 1];
%t A168175 Table[v[n][[1]], {n, 0, 30}]
%K A168175 sign,uned,new
%O A168175 0,2
%A A168175 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 19 2009

%I A168174
%S A168174 1000000001159,1000000002217,1000000003463,1000000004161,1000000005713,
%T A168174 1000000005911,1000000006037,1000000006451,1000000006699,1000000007333,
%U A168174 1000000009403,1000000010249,1000000010447,1000000010483,1000000011019
%N A168174 Emirps (A006567) with emirp number of digits and emirp digital sum.
%C A168174 Last 13-digit examples: 9999999990583, 9999999990853, 9999999995191, 9999999996901, 9999999997919, 9999999998987. First examples of some digit lengths: 10^16 + {79, 1551, 3711, 7711, 9421, 9867}; 10^30 + {2613, 29979, 37857, 41461, 47577}; 10^36 + {9061, 21081, 52351, 71017, 95781}; 10^70 + {691, 19321, 203403, 225201, 231987}; 10^72 + {97167, 158637, 227001, 233679, 265021}. ... and skipping a few other legal lengths... 10^148 + 53967 -- Jack Brennen <jfb@brennen.net>
%F A168174 {p: p in A006567 and A055642(p) in A006567 and A007953(p) in A006567}.
%e A168174 a(1) = 1000000001159, which is prime, and R(1000000001159) = 951100000000 is prime, and sod(1000000001159) = 17 is prime, and R(17) = 71 is prime.
%Y A168174 Cf. A000040, A006567, A167992, A007953, A055642, A114018, A167992.
%K A168174 base,nonn,new
%O A168174 1,1
%A A168174 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 19 2009

%I A167506
%S A167506 2,2,3,4,5,2,6,7,6,3,5,1,10,1,3,8,10,2,7,4,3,2,9,1,5,1,5,5,6,2,13,6,3,1,
%T A167506 9,5,10,2,5,7,13,1,11,6,4,0,12,1,8,3,7,9,11,1,7,7,4,2,11,1,11,2,9,6,6,1,
%U A167506 13,8,8,1,9,2,13,0,5,4,12,1,11,2,10,3,13,2,8,2,4,6,9,1,6,7,4,1,8,1,9,1
%N A167506 Number of m >= 0, m <=n such that 2^(n-m) 3^m + 1 or 2^(n-m) 3^m - 1 is prime.
%C A167506 M. Underwood observed that for all primes p < 3187 we have a(p) > 1, and asks whether there is a prime such that a(p) = 0. (This is equivalent to A167504(p) = A167505(p) = 0.)
%H A167506 M. Underwood, <a href="http://tech.groups.yahoo.com/group/primenumbers/message/21119">2^a*3^b one away from a prime</a>. Post to primenumbers group, Nov. 19, 2009.
%F A167506 maxA { A167504(n), A167505(n) } <= A167506(n) <= A167504(n)+A167505(n)
%o A167506 (PARI) A167505(n)=sum( b=0,n, ispseudoprime(3^b<<(n-b)-1) || ispseudoprime(3^b<<(n-b)+1))
%Y A167506 Cf. A167504, A167505.
%K A167506 nonn,new
%O A167506 1,1
%A A167506 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A168173
%S A168173 0,1,1,1,2,3,3,4,4,6,8,12,13,16,18,21,25,32,38,46,55,65,78,92,103,122,
%T A168173 140,165,193,229,264,305,345,395,451,517,590,682,781,893,1013,1165,1324,
%U A168173 1518,1717,1945,2188,2468,2753,3089,3457
%N A168173 Number of partitions of n in which the sum of reciprocals of parts is less than 1.
%Y A168173 Cf. A051908.
%K A168173 more,nonn,new
%O A168173 1,5
%A A168173 Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 19 2009

%I A167505
%S A167505 1,2,2,2,3,2,3,4,4,3,2,0,5,1,1,6,6,2,5,1,1,2,5,0,3,1,2,2,4,2,5,3,0,1,6,
%T A167505 2,8,2,2,3,9,1,7,4,4,0,6,0,3,3,2,7,8,1,4,4,1,2,6,0,5,2,4,2,2,1,11,4,3,1,
%U A167505 3,0,6,0,2,3,4,1,6,0,4,3,8,2,2,2,2,1,3,1,2,3,3,1,3,0,5,1,2,1,7,0,7,2,2
%N A167505 Number of primes of the form 2^(n-m) 3^m + 1, 0 <= m <= n.
%H A167505 M. Underwood, <a href="http://tech.groups.yahoo.com/group/primenumbers/message/21119">2^a*3^b one away from a prime</a>. Post to primenumbers group, Nov. 19, 2009.
%o A167505 (PARI) A167505(n)=sum(b=0,n,ispseudoprime(3^b<<(n-b)+1))
%Y A167505 Cf. A167504, A167506.
%K A167505 nonn,new
%O A167505 1,2
%A A167505 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167504
%S A167504 1,2,3,2,4,0,5,3,4,0,3,1,7,0,2,2,6,0,3,3,2,0,5,1,3,0,3,3,3,0,9,3,3,0,4,
%T A167504 3,3,0,3,4,6,0,6,2,1,0,7,1,5,0,6,2,3,0,3,3,3,0,6,1,6,0,5,4,5,0,2,4,5,0,
%U A167504 6,2,7,0,3,1,8,0,5,2,6,0,5,0,6,0,2,5,6,0,5,4,1,0,5,1,4,0,4,0,6,0,3,4,3
%N A167504 Number of primes of the form 2^(n-m) 3^m - 1, 0 <= m <= n.
%H A167504 M. Underwood, <a href="http://tech.groups.yahoo.com/group/primenumbers/message/21119">2^a*3^b one away from a prime</a>. Post to primenumbers group, Nov. 19, 2009.
%o A167504 (PARI) A167504(n)=sum(b=0,n,ispseudoprime(3^b<<(n-b)-1))
%K A167504 nonn,new
%O A167504 1,2
%A A167504 M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A168152
%S A168152 1,2,6,10,20,28,42,48,60,70,96,120,138,148,164,188,218,222,264,282,294,
%T A168152 322,348,372,412,446,460,480,490,504,582,608,636,658,710,726,762,804,
%U A168152 824,858,884,906,962,978,1004,1018,1086,1186,1206,1218,1238,1260,1282
%N A168152 Prime(prime(n)) minus prime(n).
%F A168152 a(n)=A006450(n)-A000040(n).
%Y A168152 Cf. A000040, A006450.
%K A168152 nonn,new
%O A168152 1,2
%A A168152 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 19 2009

%I A168151
%S A168151 1,2,1,6,3,1,18,9,4,1,58,29,13,5,1,192,96,44,18,6,1,650,325,151,64,24,7,
%T A168151 1,2232,1116,524,228,90,31,8,1,7746,3873,1833,813,333,123,39,9,1,27096,
%U A168151 13548,6452,2904,1222,473,164,48,10,1
%N A168151 Riordan array (1/u,(1-u)/2), u=sqrt(1-4x+4*x^3).
%C A168151 T(n,0) = A157004(n).
%F A168151 T(n,0)=2*T(n,1) for n>0, T(0,0)=1, T(n,k)=T(n-1,k-1)-T((n-3,k-1)+T(n,k+1).
%e A168151 Triangle begins : 1 ; 2,1 ; 6,3,1 ; 18,9,4,1 ; 58,29,13,5,1 ; ...
%Y A168151 Cf. A025262, A157004
%K A168151 nonn,tabl,new
%O A168151 0,2
%A A168151 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2009

%I A168009
%S A168009 1,2,3,4,5,6,9,10,11,12,21,22,23,24
%N A168009 Numbers of A168007, in sorted order.
%Y A168009 Cf. A168007, A168008.
%K A168009 more,nonn,new
%O A168009 1,2
%A A168009 Omar E. Pol (info(AT)polprimos.com), Nov 19 2009

%I A168150
%S A168150 0,1,1,3,8,20,48,112,256,576,1280,2816,6144,13312
%V A168150 0,1,-1,3,-8,20,-48,112,-256,576,-1280,2816,-6144,13312
%N A168150 Inverse binomial transform of A026741=0,1,1,3,2,5,3,7,.
%C A168150 A026741: 1) differences=1,0,2,-1,3,-2,=A028242 signed; 2) second differences:-1,2,-3,4,=A000027 signed; 3) third differences: 3,-5,7,-9=A144396 signed; 4) main diganal:0,0,-3,-9,-24,-60,=0,0,-3*A001792; 5) terms of rank 0,1,2,4,8,16,=A131577 are 0,1,1,2,4,8,16,=A166444.
%F A168150 a(n)=0,1,A001792 signed.
%Y A168150 A108189.
%K A168150 nonn,uned,new
%O A168150 0,4
%A A168150 Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009

%I A168149
%S A168149 2,6,18,24,33,34,36,40,43,67,69,77,79,91,114,119,130,153,182,187,189,
%T A168149 199,221,222,230,232,288,301,307,312,317,349,363,381,402,410,415,427,
%U A168149 444,454,465,488,504,509,511,561,573,594,629,645,647
%N A168149 Numbers n such that n^8+(n-1)^8 is a prime
%Y A168149 Cf. A153504
%K A168149 nonn,new
%O A168149 1,1
%A A168149 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 19 2009

%I A168008
%S A168008 1,2,1,3,1,5,1,3,1,11,1,3,1,23,1
%V A168008 1,2,-1,3,-1,5,-1,3,-1,11,-1,3,-1,23,-1
%N A168008 First differences of A168007.
%Y A168008 Cf. A168007, A168009.
%K A168008 more,sign,new
%O A168008 1,2
%A A168008 Omar E. Pol (info(AT)polprimos.com), Nov 19 2009

%I A168007
%S A168007 1,2,4,3,6,5,10,9,12,11,22,21,24,23
%N A168007 Jumping divisor sequence: Diagram 3, Rule 1.
%H A168007 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv09.jpg">Illustration of initial terms</a>
%H A168007 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Illustration: Periodic curves and tau(n) (Fig. 1)</a>
%H A168007 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv02.jpg">Illustration: Periodic curves and tau(n) (Fig. 2)</a>
%Y A168007 Cf. A000005, A168008, A168009.
%K A168007 hard,more,nonn,new
%O A168007 1,2
%A A168007 Omar E. Pol (info(AT)polprimos.com), Nov 19 2009

%I A168148
%S A168148 1,1,2,2,3,4,4,3,6,6,6,4,6,7,10,6
%N A168148 Row sums of triangle in A168030.
%K A168148 nonn,new
%O A168148 0,3
%A A168148 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2009

%I A168146
%S A168146 1,5,7,9,11,13,15,16,17,19,21,22,23,25,26,27,28,29,31,32,33,34,35,36,37,
%T A168146 38,39,40,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,61,62,
%U A168146 63,64,65,66,67,68,69
%N A168146 phi(n)>pi(n).
%C A168146 Numbers n such that A000010(n)> A000720(n).
%F A168146 A000010(a(n))>A000720(a(n)).
%e A168146 A000010(a(1)=1)=1>A000720(a(1)=1)=0; A000010(a(2)=5)=4>A000720(a(2)=5)=3.
%Y A168146 Cf. A000010, A000720, A037121, A037228, A073456.
%K A168146 nonn,new
%O A168146 1,2
%A A168146 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 19 2009

%I A168147
%S A168147 11,271,641,2161,33751,40961,58321,138241,196831,270001,297911,466561,
%T A168147 506531,795071,1326511,1406081,1851931,2160001,3890171,4218751,5314411,
%U A168147 5513681,6585031,7290001,8043571,11910161,12597121,12950291,14815441
%N A168147 Primes of the form p = 1 + 10*n^3 for a natural number n.
%C A168147 (1) These primes all with end digit 1=1^3 are concatenations of two CUBIC numbers: "n^3 1"
%C A168147 (2) It is conjectured that sequence is infinite
%C A168147 (3) It is an open problem if 3 consecutive naturals n exist which give such a prime
%D A168147 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A168147 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%e A168147 (1) 1+10*1^3=11 gives a(1)
%e A168147 (2) 1+10*2^3=81=3^4, 81 is no term of this sequence
%e A168147 (3) 1+10*3^3=271 gives a(2)
%Y A168147 Cf. A000040 The prime numbers
%Y A168147 Cf. A167535 Concatenation of two square numbers which give a prime
%K A168147 nonn,new
%O A168147 1,1
%A A168147 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 19 2009

%I A168145
%S A168145 1,5,6,12,18,24,36,48,60,84
%N A168145 Numbers n such that to abs(Phi[n]-Pi[n])=1.
%F A168145 Solutions to A000010(x)=A000720(x)+k where k=+-1; finite for any fixed value of k.
%e A168145 15 primes below 48=(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47); 16 terms in RRS(48)=(1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47), so 48 is here.
%Y A168145 Cf. A000010, A000720, A037121, A037228, A073456.
%K A168145 nonn,new
%O A168145 1,2
%A A168145 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 19 2009

%I A168144
%S A168144 19,23,29,43,71
%N A168144 The enlarged on 14 first differences of A168143 which are different from 1
%C A168144 All terms of the sequence are primes more than 17.
%D A168144 E. S. Rowland, A natural prime-generating recurrence , Journal of Integer Sequences, Vol.11(2008), Article 08.2.8
%H A168144 V.Shevelev, <a href="http://arXiv.org/abs/math.0910.4676">A new generator of primes based on the Rowland idea</a>
%H A168144 V.Shevelev, <a href="http://arXiv.org/abs/math.0911.3491">Generalizations of the Rowland theorem</a>
%Y A168144 A168143 A167495 A167494 A167493 A167197 A167195 A167170 A167168 A106108 A132199 A167054 A167053 A166944 A166945 A163960 A163961 A163963 A084662 A084663 A134162 A135506 A135508 A118679 A120293
%K A168144 nonn,uned,new
%O A168144 1,1
%A A168144 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 19 2009

%I A168143
%S A168143 37,38,43,44,45,46,55,56,57,58,73,74,75,76,77,78,79,80,81,82,83,84,85,
%T A168143 86,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,
%U A168143 132,133,134,135,136,137,138,139,140,141,142,199
%N A168143 a(17)=37; for n>=17, a(n)=3n-14 if gcd(n,a(n-1))>1 and all prime divisors of n more than 17; a(n)=a(n-1)+1, otherwise
%C A168143 The enlarged on 14 first differences are 15 or primes more than 17. A generalization see in our paper.
%D A168143 E. S. Rowland, A natural prime-generating recurrence , Journal of Integer Sequences, Vol.11(2008), Article 08.2.8
%H A168143 V.Shevelev, <a href="http://arXiv.org/abs/math.0910.4676">A new generator of primes based on the Rowland idea</a>
%H A168143 V.Shevelev, <a href="http://arXiv.org/abs/math.0911.3491">Generalizations of the Rowland theorem</a>
%Y A168143 A167495 A167494 A167493 A167197 A167195 A167170 A167168 A106108 A132199 A167054 A167053 A166944 A166945 A163960 A163961 A163963 A084662 A084663 A134162 A135506 A135508 A118679 A120293
%K A168143 nonn,uned,new
%O A168143 17,1
%A A168143 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 19 2009

%I A168140
%S A168140 2,3,3,7,4,7,1,7,7,1,29,31,17,31,29,1,29,16,16,29,1,43,41,104,113,104,
%T A168140 41,43,1,43,20,109,20,43,1,29,31,104,101,113,101,104,31,29,1,29,16,97,
%U A168140 109,109,97,16,29,1,71,61,172,169,1049,263,1049,169,172,61,71
%N A168140 Sum of nth numerator and nth denumerator in triangle formed from Bernoulli numbers.
%F A168140 a(n)=A085737(n)+A085738(n).
%Y A168140 Cf. A085737, A085738, A162298, A162299.
%K A168140 nonn,new
%O A168140 1,1
%A A168140 Juri-Stepan Gerasimov (2stepan(AT)ranbler.ru), Nov 19 2009

%I A168141
%S A168141 1,2,2,2,1,2,1,1,0,1,1,2,1,1,0,1,1,2,1,1,0,1,1,1,0,0,0,1,1,2,1,1,0,0,0,
%T A168141 1,1,1,0,1,1,2,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,2,1,1,0,0,0,1,1,1,0,1,
%U A168141 1,2,1,1,0,0,0,1,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0,0,1,1,1,0,1,1,2,1,1,0
%N A168141 PrimePi[n+1}-PrimePi[n-2}.
%Y A168141 Cf. A000720, A090406.
%K A168141 nonn,new
%O A168141 1,2
%A A168141 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 19 2009

%I A168142
%S A168142 2,1,8,7,6,5,4,3,2,1,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,32,31,
%T A168142 30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,
%U A168142 5,4,3,2,1,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31
%N A168142 Numeration for Janet periodic table of the elements. Extended. Janet table must be considered from right to left.
%C A168142 For Janet form in two parts: 1,2,5,6,7,8,9,10,11,12,=A138100 and 3,4,13,14,15,16,17,18,19,20,=A138101. Formulas for table in reference (1) page 15,without corresponding table;table in (2) in leaflets 2 and 3.
%D A168142 (1) JANET,Charles, La structure du Noyau de l'atome,consideree dans la Classification periodique des elements chimiques,1927 (Novembre),N. 2,BEAUVAIS,67 pages,3 leafleats. (2) JANET,charles,CONSIDERATIONS SUR LA STRUCTURE DU NOYAU DE L'ATOME,Decembre 1929,N 5,BEAUVAIS,2+45 pp.,4 leaflets.
%K A168142 nonn,uned,new
%O A168142 1,1
%A A168142 Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009

%I A168139
%S A168139 0,1,4,9,1521,1681,2025,2304,2601,3364,3481,3600,4489,4624,5776,5929,
%T A168139 7225,7396,8100,8836,9025,100000000,100020001,100040004,100060009,
%U A168139 100080016,100100025,100200100,100220121,100240144,100260169,100400400
%N A168139 Squares whose sum and number of digits are also squares.
%e A168139 1521=39^2, number of digits = 4, number of digits = 1+5+2+1=9.
%Y A168139 Subsequence of A053057 (with additional first term = 0).
%K A168139 base,nonn,new
%O A168139 0,3
%A A168139 Zak Seidov (zakseidov(AT)yahoo.com), Nov 19 2009

%I A168138
%S A168138 1,4,9,64,25,49,729,121,169,1048576,289,361,529,110075314176,15625,841,
%T A168138 656100000000,961,1369,6553600000000,1681,9682651996416,1849,2209,
%U A168138 117649,2809,72301961339136,96717311574016,3481,3721,360040606269696
%N A168138 a(n) = k^tau(n) where k is a Fibonacci number
%K A168138 nonn,new
%O A168138 1,2
%A A168138 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Nov 19 2009

%I A168137
%S A168137 1,0,8,13,14,25,24,74,112,127,125,165,265,1265,2568,12468,1002568,
%T A168137 1001002568,1000001001002568,1000000001000001001002568,
%U A168137 1001000000001000001001002568
%N A168137 a(n) = the smallest number whose American English name contains n distinct letters of the alphabet. 3<=n<=23
%C A168137 The letters j and k are unused and the letter z only appears in "zero", thus the maximum number of distinct letters possible is 23.
%C A168137 Note that in American English, it is improper to use the word "and" in the name of a number.
%C A168137 . Wrong: "one hundred and one"
%C A168137 . Right: "one hundred one"
%C A168137 Table of names for a(n)
%C A168137 ...n...name..................................letters used
%C A168137 .--------------------------------------------------------
%C A168137 ...3...one...................................one
%C A168137 ...4...zero..................................zero
%C A168137 ...5...eight.................................eight
%C A168137 ...6...thirteen..............................thiren
%C A168137 ...7...fourteen..............................fourten
%C A168137 ...8...twenty five...........................twenyfiv
%C A168137 ...9...twenty four...........................twenyfour
%C A168137 ..10...seventy four..........................sevntyfour
%C A168137 ..11...one hundred twelve....................onehudrtwlv
%C A168137 ..12...one hundred twenty seven..............onehudrtwysv
%C A168137 ..13...one hundred twenty five...............onehudrtwyfiv
%C A168137 ..14...one hundred sixty five................onehudrsixtyfv
%C A168137 ..15...two hundred sixty five................twohundresixyfv
%C A168137 ..16...one thousand two hundred
%C A168137 .........sixty five..........................onethusadwrixyfv
%C A168137 ..17...two thousand five hundred
%C A168137 .........sixty eight.........................twohusandfiverxyg
%C A168137 ..18...twelve thousand four hundred
%C A168137 .........sixty eight.........................twelvhousandfrixyg
%C A168137 ..19...one million two thousand
%C A168137 .........five hundred sixty eight............onemiltwhusadfvrxyg
%C A168137 ..20...one billion one million two
%C A168137 .........thousand five hundred sixty
%C A168137 .........eight...............................onebilmtwhusadfvrxyg
%C A168137 ..21...one quadrillion one billion
%C A168137 .........one million two thousand
%C A168137 .........five hundred sixty eight............onequadrilbmtwhsfvxyg
%C A168137 ..22...one septillion one quadrillion
%C A168137 .........one billion one million two
%C A168137 .........thousand five hundred sixty
%C A168137 .........eight...............................onesptilquadrbmwhfvxyg
%C A168137 ..23...one octillion one septillion
%C A168137 .......one quadrillion one billion
%C A168137 .......one million two thousand five
%C A168137 .......hundred sixty eight...................onectilspquadrbmwhfvxyg
%H A168137 Landon Curt Noll,<a href="http://isthe.com/chongo/tech/math/number/number.html">The English Name of a Number</a>
%H A168137 Landon Curt Noll,<a href="http://isthe.com/chongo/tech/math/number/tenpower.html">first 10000 powers of ten in the American system</a>
%e A168137 a(5) = 8, because "eight" contains five distinct letters of the alphabet.
%Y A168137 Cf. A050933, A134629
%K A168137 fini,full,nonn,word,new
%O A168137 3,3
%A A168137 Andrew Weimholt (andrew(AT)weimholt.com), Nov 19 2009

%I A168136
%S A168136 1,1,2,20,504,25200,2189088,302702400,62564261760,18427508985600,
%T A168136 7449695786856960,4010313259477324800,2803674333549374208000,
%U A168136 2492728196309155284480000,2768630339381333070099456000
%V A168136 1,1,-2,20,-504,25200,-2189088,302702400,-62564261760,18427508985600,
%W A168136 -7449695786856960,4010313259477324800,-2803674333549374208000,
%X A168136 2492728196309155284480000,-2768630339381333070099456000
%N A168136 Bernoulli(2n)*(2n+1)!/n!.
%o A168136 (PARI) a(n)=bernfrac(2*n)*(2*n+1)!/n!
%Y A168136 Cf. A001332, A004193.
%K A168136 sign,new
%O A168136 0,3
%A A168136 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 19 2009

%I A168135
%S A168135 7,11,14,18,21,22,25,28,29,32,33,35,36,39,40,42,43,44,46,47,49,50,51,53,
%T A168135 54,55,56,57,58,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,78,
%U A168135 79,80,81,82,83,85,86,87,89,90,92,93,94,96,97,100,101,103,104,107,108
%N A168135 Numbers expressible as 7*k+11*m (with non-negative k, m) exactly in one way.
%Y A168135 Cf. A168134.
%K A168135 fini,full,nonn,new
%O A168135 1,1
%A A168135 Zak Seidov (zakseidov(AT)yahoo.com), Nov 18 2009

%I A168134
%S A168134 1,2,3,4,5,6,8,9,10,12,13,15,16,17,19,20,23,24,26,27,30,31,34,37,38,41,
%T A168134 45,48,52,59
%N A168134 Numbers not of the form 7*k+11*m (with non-negative k, m).
%Y A168134 Cf. A168135.
%K A168134 fini,full,nonn,new
%O A168134 1,2
%A A168134 Zak Seidov (zakseidov(AT)yahoo.com), Nov 18 2009

%I A160868
%S A160868 0,16384,12224356333619,23360876124880896,3849782443869346726,
%T A160868 188260915444061388800,4398219345305144630169,62344855645472393641984,
%U A160868 615658186636295094703436,4622350732178621155393536
%N A160868 16384 LegendreP[17,n].
%K A160868 nonn,new
%O A160868 0,2
%A A160868 N. J. A. Sloane (njas(AT)research.att.com), Nov 19, 2009

%I A160867
%S A160867 0,1024,58349238473,45713798986752,4128627676457626,127709136544793600,
%T A160867 2058968975428834299,21362320125282024448,161120033112137396852,
%U A160867 954217375242404299776,4674429002779807245325,19650167585009522637824
%N A160867 1024 LegendreP[15,n].
%K A160867 nonn,new
%O A160867 0,2
%A A160867 N. J. A. Sloane (njas(AT)research.att.com), Nov 19, 2009

%I A160866
%S A160866 0,512,2247613027,721886012928,35730104198198,699102769400320,
%T A160866 7778198710037097,59067959750815232,340263076646454508,
%U A160866 1589596507531473408,6299974404043220015,21868102945021138432
%N A160866 512 LegendreP[13,n].
%K A160866 nonn,new
%O A160866 0,2
%A A160866 N. J. A. Sloane (njas(AT)research.att.com), Nov 19, 2009

%I A160865
%S A160865 0,128,43793863,5765980032,156401023862,1935682046080,14862118997493,
%T A160865 82608952539008,363455410347052,1339359393716352,4294566953004035,
%U A160865 12309095341172608,32166963447719778,77797775304659072
%N A160865 128 LegendreP[11,n].
%K A160865 nonn,new
%O A160865 0,2
%A A160865 N. J. A. Sloane (njas(AT)research.att.com), Nov 19, 2009

%I A160864
%S A160864 0,64,1734443,93604032,1391396086,10892513600,57713977089,234800671168,
%T A160864 789011921132,2293521500736,5949698591575,14081075036864,
%U A160864 30899647458018,63644611431232,124215678953261,231447389860800
%N A160864 64 LegendreP[9,n].
%K A160864 nonn,new
%O A160864 0,2
%A A160864 N. J. A. Sloane (njas(AT)research.att.com), Nov 19, 2009

%I A168133
%S A168133 1,4,56,99,238,3276,26820,55167,550514,3842258,16222973,56731455,
%T A168133 118396264
%N A168133 Numbers m = sum_(k=1...n) sigma(k) such that sum_(k=1...n) sigma(k) / k is integer for any k.
%C A168133 Numbers m = A024916(k) such that A024916(k) / k is integer for any k. If a(14) exists it must be bigger than 8*10^9.
%e A168133 Number a(3) = 56 = A024916(8) is in sequence because A024916(8) / 8 = 56 / 8 = 7 is integer for k = 8.
%K A168133 nonn,new
%O A168133 1,2
%A A168133 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168132
%S A168132 1,2,7,9,14,52,149,213,673,1778,3653,6831,9868
%N A168132 Numbers m = sum_(k=1...n) sigma(k) / k such that sum_(k=1...n) sigma(k) / k is integer for any k.
%C A168132 Numbers m = A024916(k) / k such that A024916(k) / k is integer for any k. If a(14) exist must be bigger than 82000.
%e A168132 Number a(3) = 7 is in sequence because A024916(8) / 8 = 56 / 8 = 7 is integer for k = 8.
%K A168132 nonn,new
%O A168132 1,2
%A A168132 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168131
%S A168131 1,2,8,11,17,63,180,259,818,2161,4441,8305,11998
%N A168131 Numbers k such that sum_(k=1...n) sigma(k) / k is integer.
%C A168131 Numbers k such that A024916(k) / k is integer. If a(14) exist must be bigger than 100000.
%e A168131 Number a(3) = 8 is in sequence because A024916(8) / 8 = 56 / 8 = 7 is integer.
%K A168131 nonn,new
%O A168131 1,2
%A A168131 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168130
%S A168130 1,8,690,109668,1235133,5475784320,6118230000
%N A168130 Numbers m = sum_(k=1...n) sigma(k) such that sum_(k=1...n) sigma(k) / sigma (k) is integer for any k.
%C A168130 Numbers m = A024916(k) such that A024916(k) / A000203(k) is integer for any k.
%e A168130 Number a(3) = 690 = A024916(29) is in sequence because A024916(29) / A000203(29) = 690 / 30 = 23 is integer for k = 29.
%K A168130 nonn,new
%O A168130 1,2
%A A168130 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168129
%S A168129 1,4,30,444,1767,86250,97920
%N A168129 Numbers m = sigma(k) such that sum_(k=1...n) sigma(k) / sigma (k) is integer for any k.
%C A168129 Numbers m = A000203(k) such that A024916(k) / A000203(k) is integer for any k.
%e A168129 Number a(3) = 30 is in sequence because A024916(29) / A000203(29) = 690 / 30 = 23 is integer for k = 29.
%K A168129 nonn,new
%O A168129 1,2
%A A168129 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168128
%S A168128 1,2,23,247,699,55921,70936
%N A168128 Numbers m = sum_(k=1...n) sigma(k) / sigma (k) such that sum_(k=1...n) sigma(k) / sigma (k) is integer for any k.
%C A168128 Numbers m = A024916(k) / A000203(k) such that A024916(k) / A000203(k) is integer for any k.
%e A168128 Number a(3) = 23 is in sequence because A024916(29) / A000203(29) = 690 / 30 = 23 is integer for k = 29.
%K A168128 nonn,new
%O A168128 1,2
%A A168128 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168127
%S A168127 1,3,29,365,1225,81595,86249
%N A168127 Numbers k such that sum_(k=1...n) sigma(k) / sigma (k) is integer.
%C A168127 Numbers k such that A024916(k) / A000203(k) is integer.
%e A168127 Number a(3) = 29 is in sequence because A024916(29) / A000203(29) = 690 / 30 = 23 is integer.
%K A168127 nonn,new
%O A168127 1,2
%A A168127 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A167507
%S A167507 4,2,4,5,6,4,3,4,4,4,3,4,5,6,8,6,5,7,7,7,5,9,9,10,11,9,8,9,9,9,6,10,10,
%T A167507 11,12,10,9,9,10,10,8,12,12,13,14,12,11,12,12,12,9,13,13,14,15,13,12,13,
%U A167507 13,13,8,12,12,13,14,12,11,12,12,12,11,12,13,14,16,14,13,15,15,15,11,15
%N A167507 Number of letters in the French spelling of the number n, not counting hyphens and spaces.
%C A167507 Sequence A007005 is a variant of this sequence, where spaces and hyphens are counted.
%H A167507 Wikitionnaire, <a href="http://fr.wiktionary.org/wiki/Annexe:Nombres_de_1_%C3%A0_100_en_fran%C3%A7ais">Annexe:Nombres de 1 a 100 en francais</a> (as of Nov. 18, 2009).
%e A167507 The terms a(0),...,a(16) represent the number of characters in the strings "zero" (where the "e" should have an accent), "un", "deux", "trois", "quatre", "cinq", "six", "sept", "huit", "neuf", "dix", "onze", "douze", "treize", "quatorze", "quinze", "seize".
%e A167507 Since spaces and punctuation are not counted, a(n) is less than the length of the character string whenever the spelling of n contains hyphens, as in "dix-sept" (a(17)=7), or spaces as in "vingt et un" (a(21)=9).
%o A167507 (PARI) A167507(n) = sum( i=1, #n=Vecsmall( French( n )), n[i]>96 )
%o A167507 /* Helper function: spell out n in French. Simplified version restricted to 0 <= n < 1000 */
%o A167507 French(n)={ n > 999 & error("n > 999 not implemented");
%o A167507 n<20 & return([ "zéro","un","deux","trois","quatre","cinq","six","sept","huit","neuf","dix","onze", "douze","treize","quatorze","quinze","seize","dix-sept","dix-huit","dix-neuf"][n+1]);
%o A167507 n >= 100 & return( Str( if( n>199, Str(French(n\100)," "), ""), "cent ", French(n%100)));
%o A167507 n > 80 & return( Str( "quatre-vingt-", French( n-80 )));
%o A167507 n%10==0 & return( Str( ["vingt","trente","quarante","cinquante","soixante", "soixante-dix","quatre-vingts"][n\10-1] ));
%o A167507 Str( French((n\10-(n>70))*10), if(n%10==1," et ","-"), French(n%10+10*(n>70)))}
%Y A167507 Cf. A005589 (English analogue), A167508 (counts distinct letters).
%K A167507 nonn,word,new
%O A167507 0,1
%A A167507 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 18 2009
%E A167507 Removed keyword "fini", added PARI code M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 19 2009

%I A167508
%S A167508 4,2,4,5,6,4,3,4,4,4,3,4,5,5,8,6,4,7,6,7,5,7,9,8,10,7,7,8,7,8,4,5,7,7,7,
%T A167508 7,7,6,7,6,7,7,9,10,7,9,10,9,9,8,8,8,10,11,9,8,10,10,9,9,8,9,10,9,11,10,
%U A167508 8,9,10,10,9,9,11,10,12,11,9,10,11,11,10,10,12,12,10,11,12,12,11,11,12
%N A167508 Number of different letters in the French spelling of the number n.
%C A167508 There is no number which can be written in French using only one letter, therefore the sequence does not contain the term 1.
%C A167508 It appears that letters "j", "k" and "w" don't occur in any number, while "m" and "l" first occur in "mille" (=1000), and "b" first occurs in "billion".
%C A167508 If an "e" with accent (as it occurs in "decillion") is considered as different from "e" without accent, then the range of the sequence should be { 2,3,...,26-3+1 }.
%H A167508 Wikitionnaire, <a href="http://fr.wiktionary.org/wiki/Annexe:Nombres_de_1_%C3%A0_100_en_fran%C3%A7ais">Annexe:Nombres de 1 a 100 en francais</a> (as of Nov. 18, 2009).
%e A167508 The terms a(0),...,a(12) represent the number of characters in the strings "zero", "un", "deux", "trois", "quatre", "cinq", "six", "sept", "huit", "neuf", "dix", "onze", "douze".
%e A167508 Since the "e" occurs twice in "treize", the number of different letters, a(13)=5, is less than the length of this string.
%e A167508 The same is true when the spelling contains hyphens as in "dix-sept" (a(17)=7) or spaces as in "vingt et un" (a(21)=9-2, since among the 9 nonblank characters, "t" and "n" occur twice).
%Y A167508 Cf. A167507.
%K A167508 fini,nonn,word,new
%O A167508 0,1
%A A167508 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 18 2009

%I A168110
%S A168110 73,97,113,12547,12611,13259,13523,14107,14563,14891
%N A168110 Palindromic primes in base 8 which are also emirps (A006567) in base 10.
%C A168110 What is a good way in OEIS to show other such pairs of bases analogous to this?
%F A168110 A029976 INTERSECTION A006567.
%e A168110 a(1) = 73 because 73 (base 8) = 111 (which is a palindrome), and R(73) = 37 which is a different prime (base 10). a(2) = 97 because 97 (base 8) = 141 (which is a palindrome), and R(97) = 79 which is a different prime (base 10). a(3) = 113 because 113 (base 8) = 161 (which is a palindrome), and R(113) = 311 which is a different prime (base 10). a(4) = 12547 because 12547 (base 8) = 30403 (which is a palindrome), and R(12547) = 74521 which is a different prime (base 10).
%Y A168110 Cf. A000040, A004086, A006567, A007094, A029976.
%K A168110 base,more,nonn,new
%O A168110 1,1
%A A168110 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 18 2009

%I A168109
%S A168109 55,66,78,91,105,120,136,153,171,190,210,231,252,273,294,315,336,357,
%T A168109 378,399,420,441,462,483,504,525,546,567,588,609,630,651,672,693,714,
%U A168109 735
%N A168109 a(n) = sum of natural numbers m such that n - 10 <= m <= n + 10.
%C A168109 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (10 + n)*(11 + n)/2 = A000217(10+n) for 0 <= n <= 10, a(n) = a(n-1) + 21 for n >= 11.
%K A168109 nonn,new
%O A168109 0,1
%A A168109 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168108
%S A168108 45,55,66,78,91,105,120,136,153,171,190,209,228,247,266,285,304,323,342,
%T A168108 361,380,399,418,437,456,475,494,513,532,551,570,589,608,627,646,665
%N A168108 a(n) = sum of natural numbers m such that n - 9 <= m <= n + 9.
%C A168108 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (9 + n)*(10 + n)/2 = A000217(9+n) for 0 <= n <= 9, a(n) = a(n-1) + 19 for n >= 10.
%K A168108 nonn,new
%O A168108 0,1
%A A168108 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168107
%S A168107 36,45,55,66,78,91,105,120,136,153,170,187,204,221,238,255,272,289,306,
%T A168107 323,340,357,374,391,408,425,442,459,476,493,510,527,544,561,578,595
%N A168107 a(n) = sum of natural numbers m such that n - 8 <= m <= n + 8.
%C A168107 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (8 + n)*(9 + n)/2 = A000217(8+n) for 0 <= n <= 8, a(n) = a(n-1) + 17 for n >= 9.
%K A168107 nonn,new
%O A168107 0,1
%A A168107 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168106
%S A168106 28,36,45,55,66,78,91,105,120,135,150,165,180,195,210,225,240,255,270,
%T A168106 285,300,315,330,345,360,375,390,405,420,435,450,465,480,495
%N A168106 a(n) = sum of natural numbers m such that n - 7 <= m <= n + 7.
%C A168106 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (7 + n)*(8 + n)/2 = A000217(7+n) for 0 <= n <= 7, a(n) = a(n-1) + 15 for n >= 8.
%K A168106 nonn,new
%O A168106 0,1
%A A168106 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A167509
%S A167509 1,6,2,3,4,17,14,22,24,53,74,92,97
%N A167509 Least positive integer written with n different letters when spelled out in French
%C A167509 There is no number which can be written in French using only one letter, therefore the sequence starts at offset n=2, cf. examples.
%C A167509 A variant of the definition would be the "least nonnegative integer ....", in which case a(4)=0 ("zero" with "accent aigu" on the "e"), all other terms remaining the same.
%C A167509 It appears that letters "j", "k" and "w" don't occur in any number, while "m" and "l" first occur in "mille" (=1000), and "b" first occurs in "billion".
%C A167509 If an "e" with accent (as it occurs in "decillion") is considered as different from "e" without accent, the sequence should have 26-3+1 terms.
%H A167509 Wikitionnaire, <a href="http://fr.wiktionary.org/wiki/Annexe:Nombres_de_1_%C3%A0_100_en_fran%C3%A7ais">Annexe:Nombres de 1 a 100 en francais</a> (as of Nov. 18, 2009).
%F A167509 a(n) = min { k | A167508(k) = n }
%e A167509 The terms a(2),...a(14) correspond to the French words un, six, deux, trois, quatre, dix-sept, quatorze, vingt-deux, vingt-quatre, cinquante-trois, soixante-quatorze, quatre-vingt-douze, quatre-vingt-dix-sept.
%e A167509 Here, "vingt-quatre" is the first term which contains a letter occurring twice, and therefore has a length greater than n; we conjecture that this is the case for all subsequent terms.
%o A167509 (PARI) A167509(n) = { for( k=1,#A167508, A167508[k]==n & return(k)); error("Found no result for n="n) }
%Y A167509 Cf. A167507, A080777, A134629.
%K A167509 fini,nonn,word,new
%O A167509 2,2
%A A167509 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 18 2009

%I A168105
%S A168105 21,28,36,45,55,66,78,91,104,117,130,143,156,169,182,195,208,221,234,
%T A168105 247,260,273,286,299,312,325,338,351,364,377,390,403,416,429,442,455,
%U A168105 468
%N A168105 a(n) = sum of natural numbers m such that n - 6 <= m <= n + 6.
%C A168105 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (6 + n)*(7 + n)/2 = A000217(6+n) for 0 <= n <= 6, a(n) = a(n-1) + 13 for n >= 7.
%K A168105 nonn,new
%O A168105 0,1
%A A168105 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168104
%S A168104 15,21,28,36,45,55,66,77,88,99,110,121,132,143,154,165,176,187,198,209,
%T A168104 220,231,242,253,264,275,286,297,308,319,330,341,352,363,374,385
%N A168104 a(n) = sum of natural numbers m such that n - 5 <= m <= n + 5.
%C A168104 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (5 + n)*(6 + n)/2 = A000217(5+n) for 0 <= n <= 5, a(n) = a(n-1) + 11 for n >= 6.
%K A168104 nonn,new
%O A168104 0,1
%A A168104 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168103
%S A168103 10,15,21,28,36,45,54,63,72,81,90,99,108,117,126,135,144,153,162,171,
%T A168103 180,189,198,207,216,225,234,243,252,261,270,279,288,297,306
%N A168103 a(n) = sum of natural numbers m such that n - 4 <= m <= n + 4.
%C A168103 a(n) = a(n-1) + 9 for n >= 5. Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (4 + n)*(5 + n)/2 = A000217(4+n) for 0 <= n <= 4, a(n) = a(n-1) + 9 for n >= 5.
%K A168103 nonn,new
%O A168103 0,1
%A A168103 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168102
%S A168102 6,10,15,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,133,140,
%T A168102 147,154,161,168,175,182,189,196,203,210,217,224
%N A168102 a(n) = sum of natural numbers m such that n - 3 <= m <= n + 3.
%C A168102 a(n) = a(n-1) + 7 for n >= 4. Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (3 + n)*(4 + n)/2 = A000217(3+n) for 0 <= n <= 3, a(n) = a(n-1) + 7 for n >= 4.
%K A168102 nonn,new
%O A168102 0,1
%A A168102 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168101
%S A168101 3,6,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,110,
%T A168101 115,120,125,130,135,140,145,150,155,160,165,170,175,180
%N A168101 a(n) = sum of natural numbers m such that n - 2 <= m <= n + 2.
%C A168101 Generalisation: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). a(n) = (2 + n)*(3 + n)/2 = A000217(2+n) for 0 <= n <= 2, a(n) = a(n-1) + 5 for n >= 3.
%K A168101 nonn,new
%O A168101 0,1
%A A168101 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009

%I A168091
%S A168091 1,16,64,441,1331,1369,9261,10201,10648,10816,68921,1002001,1030301,
%T A168091 1032256,1061208,8003241,10077696,60000516,60236288
%N A168091 Slowest increasing sequence of alternatively cubes and squares with common neighbor digits.
%C A168091 No term ending in zero allowed.
%e A168091 1 (cube),16(square); common neighbor digits = 1,
%e A168091 16(square), 64(cube): common neighbor digits = 6,
%e A168091 64(cube),441(square): common neighbor digits = 4, etc.
%Y A168091 Cf. A167994.
%K A168091 base,more,nonn,new
%O A168091 1,2
%A A168091 Zak Seidov (zakseidov(AT)yahoo.com), Nov 18 2009

%I A168090
%S A168090 1,0,2,2,0,4,4,0,8,8,0,16,16,0,32,32,0,64,64,0,128,128
%N A168090 a(n) = (1-(n mod 3) mod 2)2 ^ (rounddown(n/3) + (n mod 3) / 2 )
%D A168090 Rosen, Kenneth. Discrete Mathematics and Its Applications, McGraw-Hill Science/Engineering/Math; 6 edition (July 26, 2006), Section 4.3 Exercise 6b
%K A168090 easy,full,nonn,new
%O A168090 0,3
%A A168090 Frank Cheng (frankwcheng(AT)gmail.com), Nov 18 2009

%I A168089
%S A168089 0,0,0,0,2,2,4,16,256,65536,2147483648,2305843009213693952,
%T A168089 1329227995784915872903807060280344576,
%U A168089 110427941548649020598956093796432407239217743554726184882600387580788736
%N A168089 2^pentanacci(n)
%F A168089 CurrentTerms(n)=2^A001591(n)
%t A168089 a={1,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];z=a[[ -1]]=s;z=2^z,{n,12}],Table[0,{m,Length[a]-1}]]]
%K A168089 nonn,new
%O A168089 1,5
%A A168089 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009

%I A168088
%S A168088 0,0,0,2,2,4,16,256,32768,536870912,72057594037927936,
%T A168088 324518553658426726783156020576256,
%U A168088 411376139330301510538742295639337626245683966408394965837152256
%N A168088 2^tetranacci(n)
%F A168088 CurrentTerms(n)=2^A000078(n)
%t A168088 a={1,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];z=a[[ -1]]=s;z=2^z,{n,12}],Table[0,{m,Length[a]-1}]]]
%K A168088 nonn,new
%O A168088 1,4
%A A168088 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009

%I A168086
%S A168086 1,4,9,16,24,25,30,36,40,42,48,49,54,56,64,66,70,78,80,81,88,100,102,
%T A168086 104,105,110,112,114,120,121,128,130,132,135,136,138,140,144,148,154,
%U A168086 162,165,168,170,174,176,182,184,189,190,192,195,196,208,210,216,222
%N A168086 Numbers n such that d(n)=nonisolated number.
%C A168086 A000005 = A167759 U A168086. Where 0,1,3,5,7,8,9,10,11,13,14,15,16,17,19,20,21,22,.. are nonisolated numbers A167707. The nonisolated numbers of divisors of n. The positions of isolated numbers in A000005.
%F A168086 A000005(a(n))=nonisolated number.
%e A168086 A000005(a(1)=1)=1, A000005(a(2)=4)=3, A000005(a(3)=9)=3.
%Y A168086 Cf. A000005, A167759, A167707.
%K A168086 nonn,new
%O A168086 1,2
%A A168086 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 18 2009

%I A168087
%S A168087 1,2,3,7,4,11,12,15,16,14,27,24,47,44,127,124,147,144,224
%N A168087 a(n) = the smallest number whose Welsh name in the modern Decimal System contains n letters of the alphabet.
%C A168087 The table below shows the dervivation of the sequence:
%C A168087 ..n..Smallest.Number..Welsh.Name
%C A168087 ..2.........1.........un
%C A168087 ..3.........2.........dau.(m)./.dwy.(f)
%C A168087 ..4.........3.........tair.(f)
%C A168087 ..5.........7.........saith
%C A168087 ..6.........4.........pedwar.(m)./.pedair.(f)
%C A168087 ..7........11.........un.deg.un
%C A168087 ..8........12.........un.deg.dau
%C A168087 ..9........15.........un.deg.pump
%C A168087 .10........16.........un.deg.saith
%C A168087 .11........14.........un.deg.pedwar
%C A168087 .12........27.........dau.ddeg.saith
%C A168087 .13........24.........dau.ddeg.pedwar
%C A168087 .14........47.........pedwar.deg.saith
%C A168087 .15........44.........pedwar.deg.pedwar
%C A168087 .16.......127.........cant.dau.ddeg.saith
%C A168087 .17.......124.........cant.dau.ddeg.pedwar
%C A168087 .18.......147.........cant.pedwar.deg.saith
%C A168087 .19.......144.........cant.pedwar.deg.pedwar
%C A168087 .20.......224.........dau.gant.dau.ddeg.pedwar
%H A168087 TAKASUGI Shinji,<a href="http://www.sf.airnet.ne.jp/~ts/language/number/traditional_welsh.html">The Number System of Welsh (Modern)</a>
%K A168087 nonn,new
%O A168087 2,2
%A A168087 Christopher Hunt Gribble (chris.eveswell(AT)virgin.net), Nov 18 2009

%I A168085
%S A168085 1,2,3,7,4,12,11,13,21,16,14,27,24,19,31,33,73,36,34
%N A168085 a(n) = the smallest number whose Welsh name in the traditional Vigesimal System contains n letters of the alphabet.
%C A168085 The table below shows the derivation of the sequence:
%C A168085 ..n..Smallest.Number..Welsh.Name
%C A168085 ..2.........1.........un
%C A168085 ..3.........2.........dau.(m)./.dwy.(f)
%C A168085 ..4.........3.........tair.(f)
%C A168085 ..5.........7.........saith
%C A168085 ..6.........4.........pedwar.(m)./.pedair.(f)
%C A168085 ..7........12.........deuddeg
%C A168085 ..8........11.........un.ar.ddeg
%C A168085 ..9........13.........tri.ar.ddeg
%C A168085 .10........21.........un.ar.hugain
%C A168085 .11........16.........un.ar.bymtheg
%C A168085 .12........14.........pedwar.ar.ddeg
%C A168085 .13........27.........saith.ar.hugain
%C A168085 .14........24.........pedwar.ar.hugain
%C A168085 .15........19.........pedwar.ar.bymtheg
%C A168085 .16........31.........un.ar.ddeg.ar.hugain
%C A168085 .17........33.........tri.ar.ddeg.ar.hugain
%C A168085 .18........73.........tri.ar.ddeg.a.thrigain
%C A168085 .19........36.........un.ar.bymtheg.ar.hugain
%C A168085 .20........34.........pedwar.ar.ddeg.ar.hugain
%H A168085 TAKASUGI Shinji,<a href="http://www.sf.airnet.ne.jp/~ts/language/number/traditional_welsh.html">The Number System of Welsh (Traditional)</a>
%K A168085 nonn,new
%O A168085 2,2
%A A168085 Christopher Hunt Gribble (chris.eveswell(AT)virgin.net), Nov 18 2009

%I A168084
%S A168084 0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,
%T A168084 8191,16381,32760,65516,131024,262032,524032,1048000,2095872,4191488,
%U A168084 8382464,16763904,33525760,67047424,134086657,268156933,536281106
%N A168084 Fibonacci 13-step numbers.
%t A168084 a={1,0,0,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]
%K A168084 nonn,new
%O A168084 1,15
%A A168084 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009

%I A168083
%S A168083 0,0,0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,512,1024,2048,4095,
%T A168083 8189,16376,32748,65488,130960,261888,523712,1047296,2094336,4188160,
%U A168083 8375296,16748544,33492993,66977797,133939218,267845688,535625888
%N A168083 Fibonacci 12-step numbers.
%t A168083 a={1,0,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]
%K A168083 nonn,new
%O A168083 1,14
%A A168083 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009

%I A168082
%S A168082 0,0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,512,1024,2047,4093,8184,
%T A168082 16364,32720,65424,130816,261568,523008,1045760,2091008,4180992,8359937,
%U A168082 16715781,33423378,66830392,133628064,267190704,534250592,1068239616
%N A168082 Fibonacci 11-step numbers.
%t A168082 a={1,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]
%K A168082 nonn,new
%O A168082 1,13
%A A168082 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009

%I A168081
%S A168081 0,1,2,5,8,21,34,81,128,337,546,1301,2056,5381,8706,20737,32768,86273,
%T A168081 139778,333061,526344,1377557,2228770,5308753,8388736,22085713,35782690,
%U A168081 85262357,134742024,352649221,570556418,1359020033,2147483648
%N A168081 Lucas sequence U_n(x,1) over the field GF(2).
%C A168081 The Lucas sequence U_n(x,1) over the field GF(2)={0,1} is: 0, 1, x, x^2+1, x^3, x^4+x^2+1, x^5+x, ... Numerical values are obtained evaluating these 01-polynomials at x=2 over the integers.
%F A168081 For n>1, a(n) = (2*a(n-1)) XOR a(n-2)
%o A168081 (PARI) { a=0; b=1; for(n=1,50, c=bitxor(2*b,a); a=b; b=c; print1(c,", "); ) }
%K A168081 nonn,new
%O A168081 0,3
%A A168081 Max Alekseyev (maxale(AT)gmail.com), Nov 18 2009

%I A168079
%S A168079 1,7,13,17,19,24,25,27,29,31,33,35,36,41,43,48,49,51,52,54,55,59,61,63,
%T A168079 65,66,71,73,75,77,78,84,85,87,88,90,91,92,94,95,96,101,103,107,109,114,
%U A168079 115,116,124,125,16,132,133,135,137,139,141,142,146,147,149,151,153,155
%V A168079 -1,7,13,17,19,24,25,27,29,31,33,35,36,41,43,48,49,51,52,54,55,59,61,63,
%W A168079 65,66,71,73,75,77,78,84,85,87,88,90,91,92,94,95,96,101,103,107,109,114,
%X A168079 115,116,124,125,16,132,133,135,137,139,141,142,146,147,149,151,153,155
%N A168079 Numbers n such that exactly one of n+-1, n+-2 and n+-3 is prime.
%e A168079 a(1)=-1 (-4,-3,-2,0 and 1 are nonprimes, 2 is prime); a(2)=7 (4,6,8,9 and 10 are nonprimes, 5 is prime); a(3)=13 (8,9,10,12 and 14 are nonprimes, 11 is prime).
%Y A168079 Cf. A000027, A000040, A100317, A141468.
%K A168079 nonn,new
%O A168079 1,2
%A A168079 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 18 2009

%I A168080
%S A168080 0,1,2,3,9,12,14,15,18,21,22,26,28,30,32,34,38,39,42,45,46,50,56,60,64,
%T A168080 68,69,72,76,80,81,82,86,98,99,102,105,108,111,112,128,129,130,134,136,
%U A168080 138,140,148,150,154,160,164,165,166,170,176,178,180,190,192,195,198
%N A168080 Numbers n such that exactly two of n+-1, n+-2 and n+-3 are primes.
%e A168080 a(1)=0 (-3,-2,-1 and 1 are nonprimes, 2 and 3 are primes); a(2)=1 (-2,-1,0 and 4 are nonprimes, 2 and 3 are primes); a(3)=2 (-1,0,1 and 4 are nonprimes, 3 and 5 are primes); a(4)=3 (0,1,4 and 6 are nonprimes, 2 and 5 are primes); a(5)=9 (6,8,10 and 12 are nonprimes, 7 and 11 are primes).
%Y A168080 Cf. A000027, A000040, A100317, A141468.
%K A168080 nonn,new
%O A168080 1,3
%A A168080 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 18 2009

%I A168078
%S A168078 1,4,12,32,144,176,1728,4320,27936
%N A168078 Number of matrices with elements 1..n in which every pair of adjacent elements are relatively prime
%C A168078 For prime p, a(p) = 2*A076220(p)
%H A168078 L. Manor, <a href="a168078.gif">Illustration of terms 1-4</a>
%Y A168078 Cf. A000005, A076220
%K A168078 more,nonn,new
%O A168078 1,2
%A A168078 Lior Manor (lior.manor(AT)gmail.com), Nov 18 2009

%I A168077
%S A168077 0,1,1,9,4,25,9,49,16,81,25,121,36,169,49,225,64,289,81,361,100,441,121
%N A168077 A129194=0,1,2,9,8,25,18,49,32,; a(n)= mix A129194(2n)/2 , A129194(2n+1).
%C A168077 A000290(n+1)=Lyman denominators; A016754=first Balmer A061038 quadrisection. See submitted A168068.
%F A168077 a(n)=A026741^2. Also mix A000290 , (A000290(2n+1)=A016754) .
%K A168077 nonn,uned,new
%O A168077 0,4
%A A168077 Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2009

%I A168076
%S A168076 1,0,3,3,6,12,27,63,153,381,969,2505,6564,17394,46533,
%T A168076 125505,340902,931716,2560401,7070337,19609146,54597852,
%U A168076 152556057,427642677,1202289669,3389281245,9578183391,27130207503
%V A168076 1,0,-3,-3,-6,-12,-27,-63,-153,-381,-969,-2505,-6564,-17394,-46533,
%W A168076 -125505,-340902,-931716,-2560401,-7070337,-19609146,-54597852,
%X A168076 -152556057,-427642677,-1202289669,-3389281245,-9578183391,-27130207503
%N A168076 Expansion of 1-3*sqrt(1-x-sqrt(1-2x-3x^2))/2.
%C A168076 a(n+2)=-3*A000106(n). Hankel transform is A168075. Another variant is A168073.
%F A168076 a(n)=0^n-3*sum{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
%K A168076 easy,sign,new
%O A168076 0,3
%A A168076 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168075
%S A168075 1,3,36,189,567,2430,9477,28431,104976,373977,1121931,46943955,
%T A168075 1175547492,17473780080,193069326654,1757784154221,13934524914747,
%U A168075 99450968666463,653273852438646,4011107280777441,23282783170260237
%V A168075 1,-3,36,-189,567,-2430,9477,-28431,104976,-373977,1121931,-46943955,
%W A168075 1175547492,-17473780080,193069326654,-1757784154221,13934524914747,
%X A168075 -99450968666463,653273852438646,-4011107280777441,23282783170260237
%N A168075 Expansion of (1+27x^2-54x^3)/((1+3x)^2*(1-3x+9 x^2)).
%C A168075 Hankel transform of A168076.
%F A168075 a(n)=(-3)^n*(n^9-45n^8+846n^7-8610n^6+51345n^5-181125n^4+361584n^3-361260n^2+137264n+4480)/4480;
%F A168075 a(n)=(-3)^n*A168074(n).
%K A168075 easy,sign,new
%O A168075 0,2
%A A168075 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168074
%S A168074 1,1,4,7,7,10,13,13,16,19,19,265,2212,10960,40366,122503,323707,770101,
%T A168074 1686214,3451123,6677437,12319414,21817555,37288153,61767490,99521671,
%U A168074 156434461,240486949,362344402,536067292,779965180,1117613923
%N A168074 Expansion of (1+3x^2+2x^3)/((1-x)^2*(1+x+x^2)).
%F A168074 a(n)=(n^9-45n^8+846n^7-8610n^6+51345n^5-181125n^4+361584n^3-361260n^2+137264n+4480)/4480;
%F A168074 a(n)=A168075(n)/(-3)^n.
%K A168074 easy,nonn,new
%O A168074 0,3
%A A168074 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168073
%S A168073 1,0,3,3,6,12,27,63,153,381,969,2505,6564,17394,46533,125505,340902,
%T A168073 931716,2560401,7070337,19609146,54597852,152556057,427642677,
%U A168073 1202289669,3389281245,9578183391,27130207503,77009455428,219023318406
%N A168073 Expansion of 1+3*sqrt(1-x-sqrt(1-2x-3x^2))/2.
%C A168073 Hankel transform is A168072. a(n+2)=3*A000106(n). Another variant is A168076.
%F A168073 a(n)=0^n+3*sum{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
%Y A168073 Cf. A168055, A168049.
%K A168073 easy,nonn,new
%O A168073 0,3
%A A168073 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168072
%S A168072 1,3,18,135,405,1944,8019,24057,91854,334611,1003833,46589661,
%T A168072 1174484610,17470591434,193059760716,1757755456407,13934438821305,
%U A168072 99450710386137,653273077597668,4011104956254507,23282776196691435
%V A168072 1,3,-18,-135,-405,-1944,-8019,-24057,-91854,-334611,-1003833,-46589661,
%W A168072 -1174484610,-17470591434,-193059760716,-1757755456407,-13934438821305,
%X A168072 -99450710386137,-653273077597668,-4011104956254507,-23282776196691435
%N A168072 Expansion of (1-27x^2-108x^3)/((1-3x)^2*(1+3x+9x^2)).
%C A168072 Hankel transform of A168073.
%F A168072 a(n)=3^n*A168071(n).
%K A168072 easy,sign,new
%O A168072 0,2
%A A168072 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168071
%S A168071 1,1,2,5,5,8,11,11,14,17,17,263,2210,10958,40364,122501,
%T A168071 323705,770099,1686212,3451121,6677435,12319412,21817553,
%U A168071 37288151,61767488,99521669,156434459,240486947,362344400
%V A168071 1,1,-2,-5,-5,-8,-11,-11,-14,-17,-17,-263,-2210,-10958,-40364,-122501,
%W A168071 -323705,-770099,-1686212,-3451121,-6677435,-12319412,-21817553,
%X A168071 -37288151,-61767488,-99521669,-156434459,-240486947,-362344400
%N A168071 Expansion of (1-3x^2-4x^3)/((1-x)^2*(1+x+x^2)).
%F A168071 a(n)=-(n^9-45n^8+846n^7-8610n^6+51345n^5-181125n^4+361584n^3-361260n^2+137264n-4480)/4480;
%F A168071 a(n)=A168072(n)/3^n.
%Y A168071 Cf. A168053.
%K A168071 easy,sign,new
%O A168071 0,3
%A A168071 Paul Barry (pbarry(AT)wit.ie), Nov 18 2009

%I A168070
%S A168070 0,1,10,100,1000,10000,100000,1000000,10000000,71624305,100000000,
%T A168070 103849576,105823694,106597243,108326947,120463578,124093657,126509743,
%U A168070 129306745,129738560,139784256,140786329,147863502,148936025,150973624
%N A168070 Numbers n with property that n, n^2, n^3, and n^4 have the same set of digits (not counting repetitions).
%C A168070 If n is here then also 10*n is. But not vice versa: if the term n is multiple of 10 then not necessarily n/10 is the term; e.g., 129738560, 172836950, 175438290 are terms but 12973856, 17283695, 17543829 not.
%Y A168070 Cf. A029800.
%K A168070 base,nonn,new
%O A168070 0,3
%A A168070 Zak Seidov (zakseidov(AT)yahoo.com), Nov 18 2009

%I A168069
%S A168069 1,2,3,2,1,3,8,2,4,6,13,3,9,8,6,16,11,4,21,10,8,13,19,5,14,9,22,16,69,6,
%T A168069 24,18,25,11,7,29,21,12,17
%N A168069 a(n) is the index k of the smallest k-th prime p(k) with f(k,n):=(p(k) + p(k+1))/n an integer (n = 1,2,3,...)
%C A168069 (1) EVERY natural k appears in the sequence, some more than once
%C A168069 (2) Theoretical interest for cases (I) a(n) < n, (II) a(n) = n, (I) a(n) > n
%C A168069 (3) Note cases n=1, 3, 16, ... with a(n) = n
%D A168069 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A168069 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%e A168069 (1) p(1)+p(2)=2+3=1 x 5 gives a(1)=a(5)=1
%e A168069 (2) p(3)+p(4)=5+7=2^2 x 3 gives a(3)=a(6)=a(12)=3, but a(2)=2 < 3, because p(2)+p(2)=2 x 2^2
%e A168069 (3) p(16)+p(17)=53+59=2^4 x 7=16 x 7 gives a(16)=16
%e A168069 (4) p(69)+p(70)=347+349=2^3 x 3 x 29 gives a(29)=69
%Y A168069 A000040 The prime numbers
%Y A168069 A167790
%K A168069 nonn,new
%O A168069 1,2
%A A168069 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 18 2009

%I A168068
%S A168068 0,0,1,0,1,1,0,1,2,3,0,1,4,3,2,0,1,8,3,4,5,0,1,16,3,8,5,3,0,1,32,3,16,5,
%T A168068 6,7,0,1,64,3,32,5,12,7,4
%N A168068 Family linked to Rydberg-Ritz spectra of hydrogen atom. Antidiagonal writing of array of successive mix A026741(2n)*2^n , A026741(2n+1).
%C A168068 Rows are 0,1,1,3,2,5,3,7,4,=A026741 (1) ; 0,1,2,3,4,5,6,7,8,=A001477 (2) ; 0,1,4,3,8,5,12,7,16,=A022998 (3) ; 0,1,8,3,16,5,24,7,32,=A144433(n-2) (4); 0,1,16,3,32,5,48,7,64, (5) ; 0,1,32,3,64,5,96,7,128, (6) ; For (1) see A026741 (Curtzz is incorrect). For (2):A001477^2=A000290.A000290(n+1)=Lyman denominators. For (3) see A154615, A145979 (from Balmer A061038),first bisection. For (4) see A106833 and A152977.
%K A168068 nonn,uned,new
%O A168068 0,9
%A A168068 Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2009

%I A166994
%S A166994 1,2,3,3,8,55,4,15,216,43631,5,24,567,318464,99515655135,6,35,1216,
%T A166994 1475631,2175583184000,4723258824886629604131775,7,48,2295,5264000,
%U A166994 27707792335839,767711852760361479511965696
%N A166994 Triangle, read by rows, where T(n,k) = T(n,k-1)^2 - T(k-1,k-1)^2 for n>=k>1, with T(n,1) = n for n>=1.
%F A166994 Main diagonal is A083869, which obeys an interesting recursion of nested radicals.
%e A166994 Triangle begins:
%e A166994 1;
%e A166994 2,3;
%e A166994 3,8,55;
%e A166994 4,15,216,43631;
%e A166994 5,24,567,318464,99515655135;
%e A166994 6,35,1216,1475631,2175583184000,4723258824886629604131775;
%e A166994 7,48,2295,5264000,27707792335839,767711852760361479511965696,589359179694820074404152604620573424809709490316113791; ...
%e A166994 ILLUSTRATE THE RECURRENCE.
%e A166994 For row 4, start with 4, then continue with the rule:
%e A166994 "obtain the next term in the row by squaring the current term and subtracting the square of the first term in the current column":
%e A166994 4^2 - 1^2 = 15; 15^2 - 3^2 = 216; 216^2 - 55^2 = 43631.
%e A166994 Likewise for row 5:
%e A166994 5^2 - 1^2 = 24; 24^2 - 3^2 = 567; 567^2 - 55^2 = 318464; 318464^2 - 43631^2 = 99515655135.
%e A166994 Continuing in this way generates all rows of this triangle.
%e A166994 ILLUSTRATE GENERATING METHOD USING NESTED RADICALS.
%e A166994 Let a(n) = A083869(n), then row n equals the resulting integers at each stage in the successive nested radicals:
%e A166994 sqrt(a(1)^2+sqrt(a(2)^2+sqrt(a(3)^2+(....+sqrt(a(n)^2)))...).
%e A166994 For example, the terms in row n=3 are:
%e A166994 3 = sqrt(1^2 + sqrt(3^2 + sqrt(55^2))),
%e A166994 8 = sqrt(3^2 + sqrt(55^2)),
%e A166994 55 = sqrt(55^2).
%e A166994 And the terms in row 4 are:
%e A166994 4 = sqrt(1^2 + sqrt(3^2 + sqrt(55^2 + sqrt(43631^2)))),
%e A166994 15 = sqrt(3^2 + sqrt(55^2 + sqrt(43631^2))),
%e A166994 216 = sqrt(55^2 + sqrt(43631^2)),
%e A166994 43631 = sqrt(43631^2).
%o A166994 (PARI) T(n,k)=if(k==1,n,T(n,k-1)^2-T(k-1,k-1)^2)
%Y A166994 Cf. A083869.
%K A166994 nonn,tabl,new
%O A166994 1,2
%A A166994 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 18 2009

%I A168066
%S A168066 0,1,1,4,1,5,1,13,6,7,1,17,1,9,8,40,1,22,1,25,10,13,1,53,10,15,28,33,1,
%T A168066 32,1,121,14,19,12,70,1,21,16,79,1,42,1,49,40,25,1,161,14,46,20,57,1,92,
%U A168066 16,105,22,31,1,104,1,33,52,364,18,62,1,73,26,60,1,214,1,39,56,81,18,72
%N A168066 If n = Product p(k)^e(k) then a(n) = {Product (p(k)+1)^e(k) - Product (p(k)-1)^e(k)}/2, a(1) = 0.
%C A168066 a(n) = 0 iff n is 1;
%C A168066 a(n) = 1 iff n prime p;
%C A168066 a(n) = p+q iff n is biprime, i.e. n = pq, p <= q primes;
%C A168066 a(n) = (pq+pr+qr)+1 iff n is triprime, i.e. n = pqr, p <= q <= r primes;
%C A168066 a(n) = (pqr+pqs+prs+qrs)+(p+q+r+s) iff n is quadprime, i.e. n = pqrs, p <= q <= r <= s primes;
%C A168066 ...
%F A168066 a(n) = {A003959(n) - A003958(n)}/2
%Y A168066 Cf. A003958, A003959, A168065.
%K A168066 nonn,new
%O A168066 1,4
%A A168066 Daniel Forgues (squid(AT)zensearch.com), Nov 18 2009

%I A168065
%S A168065 1,2,3,5,5,7,7,14,10,11,11,19,13,15,16,41,17,26,19,29,22,23,23,55,26,27,
%T A168065 36,39,29,40,31,122,34,35,36,74,37,39,40,83,41,54,43,59,56,47,47,163,50,
%U A168065 62,52,69,53,100,56,111,58,59,59,112,61,63,76,365,66,82,67,89,70,84,71
%N A168065 If n = Product p(k)^e(k) then a(n) = {Product (p(k)+1)^e(k) + Product (p(k)-1)^e(k)}/2, a(1) = 1.
%C A168065 a(n) = n iff n is 1 or prime p;
%C A168065 a(n) = n+1 iff n is biprime, i.e. n = pq, p <= q primes;
%C A168065 a(n) = n+(p+q+r) iff n is triprime, i.e. n = pqr, p <= q <= r primes;
%C A168065 a(n) = n+(pq+pr+ps+qr+qs+rs)+1 iff n is quadprime, i.e. n = pqrs, p <= q <= r <= s primes;
%C A168065 ...
%F A168065 a(n) = {A003959(n) + A003958(n)}/2
%Y A168065 Cf. A003958, A003959, A168066.
%K A168065 nonn,new
%O A168065 1,2
%A A168065 Daniel Forgues (squid(AT)zensearch.com), Nov 18 2009

%I A168063
%S A168063 1,3,5,6,9,12,15,18,21,30,39,42,45,60,69,72,81,99,102,108,111,129,138,
%T A168063 150,165,180,192,195,198,225,228,231,240,270,282,309,312,315,348,351,
%U A168063 381,399,420,432,441,459,462,465,489,501,522,570,600,609,615,618,640
%N A168063 Nnumbers n such that exactly two of n+-1 and n+-2 are primes.
%e A168063 a(1)=1 (-1 and 0 are nonprimes, 2 and 3 are primes); a(2)=3 (0 and 4 are nonprimes, 2 and 5 are primes); a(3)=5 (4 and 6 are nonprimes, 3 and 7 are primes); a(4)=6 (4 and 8 are nonprimes, 5 and 7 are primes); a(5)=9 (8 and 10 are nonprimes, 7 and 11 are primes).
%Y A168063 Cf. A000027, A000040, A100317, A141468.
%K A168063 nonn,new
%O A168063 1,2
%A A168063 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 18 2009

%I A168064
%S A168064 0,2,7,8,10,11,13,14,16,17,19,20,22,24,25,27,28,29,31,32,33,35,36,38,40,
%T A168064 41,43,44,46,48,49,51,52,54,55,57,58,59,61,62,63,65,66,68,70,71,73,74,
%U A168064 75,77,78,80,82,84,85,87,88,90,91,95,96,98,100,101,103,104,106,107,109
%N A168064 Numbers n such that exactly one of n+-1 and n+-2 is prime.
%e A168064 a(1)=0 (-2,-1 and 1 are nonprime, 2 is prime); a(2)=2 (0,1 and 4 are nonprime, 3 is prime); a(3)=7 (6,8 and 9 are nonprime, 5 is prime); a(4)=10 (8,9 and 12 are nonprime, 11 is prime); a(5)=11 (9,10 and 12 are nonprime, 13 is prime).
%Y A168064 Cf. A000027, A000040, A100317, A141468.
%K A168064 nonn,new
%O A168064 1,2
%A A168064 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 18 2009

%I A167526
%S A167526 2,7,73,293339
%N A167526 Prime factors of the speed of light in a vacuum, c = 299792458 (m/s).
%C A167526 Note that each prime factor of 299792458 appears with multiplicity 1 in this number.
%C A167526 Giving the speed of light in km/s or cm/s would yield the same prime factors > 5. If time is measured in hours, or distance in miles, feet, inches or similar, then the numerical value of c is no more an integer.
%F A167526 299792458 = 2 * 7 * 73 * 293339.
%o A167526 (PARI) factor(299 792 458)~[1,]
%Y A167526 Cf. A003678 (speed of light), A132997 (all divisors of c, contains this as subsequence).
%K A167526 fini,full,nonn,new
%O A167526 1,1
%A A167526 M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 18 2009

%I A166993
%S A166993 1,1,5,32,266,2499,25765,283084,3264502,39077898,481942608,6089941550,
%T A166993 78523226064,1029859481949,13704960309415,184688556173542,
%U A166993 2516342539576510,34617557176739174,480336524752492608
%N A166993 G.f.: A(x) = exp( Sum_{n>=1} A005260(n)/2 * x^n/n ) where A005260(n) = Sum_{k=0..n} C(n,k)^4.
%F A166993 Self-convolution yields A166992.
%e A166993 G.f.: A(x) = 1 + x + 5*x^2 + 32*x^3 + 266*x^4 + 2499*x^5 + 25765*x^6 +...
%e A166993 log(A(x)) = x + 9*x^2/2 + 82*x^3/3 + 905*x^4/4 + 10626*x^5/5 + 131922*x^6/6 + 1697508*x^7/7 +...+ A005260(n)/2*x^n/n +...
%o A166993 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^4)/2*x^m/m)+x*O(x^n)),n)}
%Y A166993 Cf. A005260, A166991, A166992.
%K A166993 nonn,new
%O A166993 0,3
%A A166993 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A166992
%S A166992 1,2,11,74,621,5850,60212,659712,7583514,90494068,1112755389,
%T A166992 14022849582,180362150901,2360201899690,31344689243344,421621652965160,
%U A166992 5734850816825046,78773961705345324,1091497852618784390
%N A166992 G.f.: A(x) = exp( Sum_{n>=1} A005260(n)*x^n/n ) where A005260(n) = Sum_{k=0..n} C(n,k)^4.
%F A166992 Self-convolution of A166993.
%e A166992 G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 621*x^4 + 5850*x^5 + 60212*x^6 +...
%e A166992 log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 1810*x^4/4 + 21252*x^5/5 + 263844*x^6/6 + 3395016*x^7/7 +...+ A005260(n)*x^n/n +...
%o A166992 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^4)*x^m/m)+x*O(x^n)),n)}
%Y A166992 Cf. A005260, A166990, A166993.
%K A166992 nonn,new
%O A166992 0,2
%A A166992 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A166991
%S A166991 1,1,3,12,57,300,1693,10045,61890,392688,2550843,16891566,113660475,
%T A166991 775223595,5349057132,37280705406,262119009927,1857241951359,
%U A166991 13250054817027,95110710932424,686490953423700,4979704242810870
%N A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)/2 * x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
%F A166991 Self-convolution yields A166990.
%e A166991 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
%e A166991 log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
%o A166991 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)/2*x^m/m)+x*O(x^n)),n)}
%Y A166991 Cf. A000172 (Franel numbers), A166990.
%K A166991 nonn,new
%O A166991 0,3
%A A166991 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A166990
%S A166990 1,2,7,30,147,786,4472,26644,164477,1044258,6782484,44887236,301782361,
%T A166990 2056250570,14172792355,98667874038,692948001906,4904403499992,
%U A166990 34951124337300,250617829087656,1807055528439771,13095146839953030
%N A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
%C A166990 Analogous to the square of the g.f. of Catalan numbers (A000108):
%C A166990 C(x)^2 = exp( Sum_{n>=1} A000984(n)*x^n/n ) where central binomial coefficient A000984(n) = Sum_{k=0..n} C(n,k)^2.
%F A166990 Self-convolution of A166991.
%e A166990 G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
%e A166990 log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
%o A166990 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}
%Y A166990 Cf. A000172 (Franel numbers), A166991.
%K A166990 nonn,new
%O A166990 0,2
%A A166990 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A160863
%S A160863 1,154,2199,13911,57209,179988,471675,1082509,2246545,4308382,7753615,
%T A160863 13243011,21650409,34104344,52033395,77215257,111829537,158514274,
%U A160863 220426183,301304623,405539289,538241628,705319979,913558437,1170699441
%N A160863 G.f.: (1+147*x+1142*x^2+1717*x^3+656*x^4+60*x^5+x^6)/(1-x)^7.
%C A160863 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160863 nonn,new
%O A160863 0,2
%A A160863 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160854
%S A160854 1,154,2155,13524,55400,173911,455120,1043547,2164267,4148584,7463281,
%T A160854 12743446,20828874,32804045,50041678,74249861,107522757,152394886,
%U A160854 211898983,289627432,389797276,517318803,677867708,877960831,1125035471
%N A160854 G.f.: (1+147*x+1098*x^2+1638*x^3+632*x^4+59*x^5+x^6)/(1-x)^7.
%C A160854 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160854 nonn,new
%O A160854 0,2
%A A160854 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160853
%S A160853 1,154,2287,14735,61227,193897,510420,1175273,2445121,4698328,8468593,
%T A160853 14482711,23702459,37370607,57061054,84733089,122789777,174140470,
%U A160853 242267443,331296655,446072635,592237493,776314056,1005793129
%N A160853 G.f.: (1+147*x+1230*x^2+1925*x^3+754*x^4+67*x^5+x^6)/(1-x)^7.
%C A160853 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160853 nonn,new
%O A160853 0,2
%A A160853 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160841
%S A160841 1,154,2287,14725,61147,193546,509293,1172305,2438317,4684258,8441731,
%T A160841 14434597,23620663,37237474,56852209,84415681,122320441,173462986,
%U A160841 241310071,329969125,444262771,589807450,773096149,1001585233
%N A160841 G.f.: (1+147*x+1230*x^2+1915*x^3+744*x^4+66*x^5+x^6)/(1-x)^7.
%C A160841 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160841 nonn,new
%O A160841 0,2
%A A160841 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160840
%S A160840 1,154,2287,14695,60907,192493,505912,1163401,2417905,4642048,8361145,
%T A160840 14290255,23375275,36838075,56225674,83463457,120912433,171430534,
%U A160840 238437955,325986535,438833179,582517321,763442428,988961545,1267466881
%N A160840 G.f.: (1+147*x+1230*x^2+1885*x^3+714*x^4+63*x^5+x^6)/(1-x)^7.
%C A160840 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160840 nonn,new
%O A160840 0,2
%A A160840 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160839
%S A160839 78,1662,13488,65481,231486,660921,1619353,3537997,7072138,13168476,
%T A160839 23141394,38758149,62332986,96830175,145975971,214379497,307662550,
%U A160839 432598330,597259092,811172721,1085488230,1433150181,1869082029
%N A160839 G.f.: (78+1116*x+3492*x^2+3237*x^3+927*x^4+72*x^5+x^6)/(1-x)^7.
%C A160839 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160839 nonn,new
%O A160839 0,1
%A A160839 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160838
%S A160838 1,45,557,3473,14417,45965,121997,283137,593281,1147213,2079309,3573329,
%T A160838 5873297,9295469,14241389,21212033,30823041,43821037,61101037,83724945,
%U A160838 112941137,150205133,197201357,255865985,328410881,417348621,525518605
%N A160838 G.f.: (1+38*x+263*x^2+484*x^3+263*x^4+38*x^5+x^6)/(1-x)^7.
%C A160838 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160838 nonn,new
%O A160838 0,2
%A A160838 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160837
%S A160837 1,45,556,3457,14317,45565,120772,280001,586225,1132813,2052084,3524929,
%T A160837 5791501,9162973,14034364,20898433,30360641,43155181,60162076,82425345,
%U A160837 111172237,147833533,194064916,251769409,323120881,410588621,516962980
%N A160837 G.f.: (1+38*x+262*x^2+475*x^3+254*x^4+37*x^5+x^6)/(1-x)^7.
%C A160837 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160837 nonn,new
%O A160837 0,2
%A A160837 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160836
%S A160836 1,69,1027,6825,29073,93789,250363,582737,1222801,2366005,4289187,
%T A160836 7370617,12112257,19164237,29351547,43702945,63482081,90220837,
%U A160836 125754883,172261449,232299313,308851005,405367227,525813489,674718961
%N A160836 G.f.: (1+62*x+565*x^2+1050*x^3+485*x^4+52*x^5+x^6)/(1-x)^7.
%C A160836 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160836 nonn,new
%O A160836 0,2
%A A160836 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160835
%S A160835 1,51,675,4319,18131,58121,154701,359605,754189,1459111,2645391,4546851,
%T A160835 7473935,11828909,18122441,26991561,39219001,55753915,77733979,
%U A160835 106508871,143665131,191052401,250811045,325401149,417632901,530698351
%N A160835 G.f.: (1+44*x+339*x^2+630*x^3+323*x^4+42*x^5+x^6)/(1-x)^7.
%C A160835 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160835 nonn,new
%O A160835 0,2
%A A160835 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160834
%S A160834 1,69,1029,6857,29273,94589,252813,589009,1236913,2394805,4343637,
%T A160834 7467417,12275849,19429229,29765597,44330145,64406881,91552549,
%U A160834 127632805,174860649,235837113,313594205,411640109,534006641,685298961
%N A160834 G.f.: (1+62*x+567*x^2+1068*x^3+503*x^4+54*x^5+x^6)/(1-x)^7.
%C A160834 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160834 nonn,new
%O A160834 0,2
%A A160834 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160833
%S A160833 1,69,1031,6889,29473,95389,255263,595281,1251025,2423605,4398087,
%T A160833 7564217,12439441,19694221,30179647,44957345,65331681,92884261,
%U A160833 129510727,177459849,239374913,318337405,417912991,542199793,695878961
%N A160833 G.f.: (1+62*x+569*x^2+1086*x^3+521*x^4+56*x^5+x^6)/(1-x)^7.
%C A160833 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160833 nonn,new
%O A160833 0,2
%A A160833 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160832
%S A160832 1,69,1026,6809,28973,93389,249138,579601,1215745,2351605,4261962,
%T A160832 7322217,12030461,19031741,29144522,43389345,63019681,89554981,
%U A160832 124815922,170961849,230530413,306479405,402230786,521716913,669428961
%N A160832 G.f.: (1+62*x+564*x^2+1041*x^3+476*x^4+51*x^5+x^6)/(1-x)^7.
%C A160832 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160832 nonn,new
%O A160832 0,2
%A A160832 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160831
%S A160831 1,69,1032,6905,29573,95789,256488,598417,1258081,2438005,4425312,
%T A160831 7612617,12521237,19826717,30386672,45270945,65794081,93550117,
%U A160831 130449688,178759449,241143813,320709005,421049432,546296369,701168961
%N A160831 G.f.: (1+62*x+570*x^2+1095*x^3+530*x^4+57*x^5+x^6)/(1-x)^7.
%C A160831 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160831 nonn,new
%O A160831 0,2
%A A160831 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160829
%S A160829 1,51,673,4287,17931,57321,152251,353333,740077,1430311,2590941,4450051,
%T A160829 7310343,11563917,17708391,26364361,38294201,54422203,75856057,
%U A160829 103909671,140127331,186309201,244538163,317207997,407052901,517178351
%N A160829 G.f.: (1+44*x+337*x^2+612*x^3+305*x^4+40*x^5+x^6)/(1-x)^7.
%C A160829 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160829 nonn,new
%O A160829 0,2
%A A160829 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160817
%S A160817 1,69,1026,6809,28973,93389,249138,579601,1215745,2351605,4261962,
%T A160817 7322217,12030461,19031741,29144522,43389345,63019681,89554981,
%U A160817 124815922,170961849,230530413,306479405,402230786,521716913,669428961
%N A160817 G.f.: (1+62*x+564*x^2+1041*x^3+476*x^4+51*x^5+x^6)/(1-x)^7.
%C A160817 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160817 nonn,new
%O A160817 0,2
%A A160817 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160816
%S A160816 1,69,1025,6793,28873,92989,247913,576465,1208689,2337205,4234737,
%T A160816 7273817,11948665,18899245,28937497,43075745,62557281,88889125,
%U A160816 123876961,169662249,228761513,304107805,399094345,517620337,664138961
%N A160816 G.f.: (1+62*x+563*x^2+1032*x^3+467*x^4+50*x^5+x^6)/(1-x)^7.
%C A160816 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160816 nonn,new
%O A160816 0,2
%A A160816 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160815
%S A160815 1,69,1024,6777,28773,92589,246688,573329,1201633,2322805,4207512,
%T A160815 7225417,11866869,18766749,28730472,42762145,62094881,88223269,
%U A160815 122938000,168362649,226992613,301736205,395957904,513523761,658848961
%N A160815 G.f.: (1+62*x+562*x^2+1023*x^3+458*x^4+49*x^5+x^6)/(1-x)^7.
%C A160815 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160815 nonn,new
%O A160815 0,2
%A A160815 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160788
%S A160788 1,69,1023,6761,28673,92189,245463,570193,1194577,2308405,4180287,
%T A160788 7177017,11785073,18634253,28523447,42448545,61632481,87557413,
%U A160788 121999039,167063049,225223713,299364605,392821463,509427185,653558961
%N A160788 G.f.: (1+62*x+561*x^2+1014*x^3+449*x^4+48*x^5+x^6)/(1-x)^7.
%C A160788 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160788 nonn,new
%O A160788 0,2
%A A160788 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160787
%S A160787 21,209,938,2833,6771,13881,25544,43393,69313,105441,154166,218129,
%T A160787 300223,403593,531636,688001,876589,1101553,1367298,1678481,2040011,
%U A160787 2457049,2935008,3479553,4096601,4792321,5573134,6445713,7416983
%N A160787 G.f.: (21+104*x+103*x^2+23*x^3+x^4)/(1-x)^5.
%C A160787 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160787 nonn,new
%O A160787 0,1
%A A160787 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160769
%S A160769 21,206,917,2757,6571,13446,24711,41937,66937,101766,148721,210341,
%T A160769 289407,388942,512211,662721,844221,1060702,1316397,1615781,1963571,
%U A160769 2364726,2824447,3348177,3941601,4610646,5361481,6200517,7134407
%N A160769 G.f.: (21+101*x+97*x^2+22*x^3+x^4)/(1-x)^5.
%C A160769 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160769 nonn,new
%O A160769 0,1
%A A160769 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160768
%S A160768 21,203,896,2681,6371,13011,23878,40481,64561,98091,143276,202553,
%T A160768 278591,374291,492786,637441,811853,1019851,1265496,1553081,1887131,
%U A160768 2272403,2713886,3216801,3786601,4428971,5149828,5955321,6851831
%N A160768 G.f.: (21+98*x+91*x^2+21*x^3+x^4)/(1-x)^5.
%C A160768 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160768 nonn,new
%O A160768 0,1
%A A160768 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160767
%S A160767 1,17,103,367,971,2131,4117,7253,11917,18541,27611,39667,55303,75167,
%T A160767 99961,130441,167417,211753,264367,326231,398371,481867,577853,687517,
%U A160767 812101,952901,1111267,1288603,1486367,1706071,1949281,2217617,2512753
%N A160767 G.f.: (1+12*x+28*x^2+12*x^3+x^4)/(1-x)^5.
%C A160767 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160767 nonn,new
%O A160767 0,2
%A A160767 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160766
%S A160766 1,19,121,439,1171,2581,4999,8821,14509,22591,33661,48379,67471,91729,
%T A160766 122011,159241,204409,258571,322849,398431,486571,588589,705871,839869,
%U A160766 992101,1164151,1357669,1574371,1816039,2084521,2381731,2709649,3070321
%N A160766 G.f.: (1+14*x+36*x^2+14*x^3+x^4)/(1-x)^5.
%C A160766 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160766 nonn,new
%O A160766 0,2
%A A160766 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160765
%S A160765 1,18,112,403,1071,2356,4558,8037,13213,20566,30636,44023,61387,83448,
%T A160765 110986,144841,185913,235162,293608,362331,442471,535228,641862,763693,
%U A160765 902101,1058526,1234468,1431487,1651203,1895296,2165506,2463633,2791537
%N A160765 G.f.: (1+13*x+32*x^2+13*x^3+x^4)/(1-x)^5.
%C A160765 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160765 nonn,new
%O A160765 0,2
%A A160765 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160753
%S A160753 1,17,103,367,971,2131,4117,7253,11917,18541,27611,39667,55303,75167,
%T A160753 99961,130441,167417,211753,264367,326231,398371,481867,577853,687517,
%U A160753 812101,952901,1111267,1288603,1486367,1706071,1949281,2217617,2512753
%N A160753 G.f.: (1+12*x+28*x^2+12*x^3+x^4)/(1-x)^5.
%C A160753 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160753 nonn,new
%O A160753 0,2
%A A160753 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160750
%S A160750 1,16,94,331,880,1951,3811,6784,11251,17650,26476,38281,53674,73321,
%T A160750 97945,128326,165301,209764,262666,325015,397876,482371,579679,691036,
%U A160750 817735,961126,1122616,1303669,1505806,1730605,1979701,2254786,2557609
%N A160750 G.f.: (1+11*x+24*x^2+11*x^3+10*x^4)/(1-x)^5.
%C A160750 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160750 nonn,new
%O A160750 0,2
%A A160750 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160749
%S A160749 5,20,46,83,131,190,260,341,433,536,650,775,911,1058,1216,1385,1565,
%T A160749 1756,1958,2171,2395,2630,2876,3133,3401,3680,3970,4271,4583,4906,5240,
%U A160749 5585,5941,6308,6686,7075,7475,7886,8308,8741,9185,9640,10106,10583
%N A160749 G.f.: (5+5*x+x^2)/(1-x)^3.
%C A160749 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160749 nonn,new
%O A160749 0,1
%A A160749 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160748
%S A160748 1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,
%T A160748 841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,2245,
%U A160748 2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325,4513
%N A160748 G.f.: (1+2*x+x^2)/(1-x)^3.
%C A160748 Source: the De Loera et al. article and the Haws website listed in A160747.
%K A160748 nonn,new
%O A160748 0,2
%A A160748 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A160747
%S A160747 1,15,85,295,771,1681,3235,5685,9325,14491,21561,30955,43135,58605,77911,
%T A160747 101641,130425,164935,205885,254031,310171,375145,449835,535165,632101,
%U A160747 741651,864865,1002835,1156695,1327621,1516831,1725585,1955185,2206975
%N A160747 G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^5.
%C A160747 Ehrhart series for matroid K_4.
%D A160747 J. A. De Loera, D. C. Haws and M. Koppe, Ehrhart Polynomials of Matroid Polytopes and Polymatroids, Discrete Comput. Geom., 42 (2009), 670-702.
%H A160747 D. C. Haws, <a href="http://www.math.ucdavis.edu/~haws/Matroids/">Matroids</a>
%K A160747 nonn,new
%O A160747 0,2
%A A160747 N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2009

%I A167007
%S A167007 1,2,5,26,501,42262,14564184,18926665052,96371663657380,
%T A167007 1825266130738144920,136764680697906838980633,
%U A167007 38133043109557952095731186822,42464330390232136488003531922964743
%N A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} C(n,k)^n.
%e A167007 G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 501*x^4 + 42262*x^5 +...
%e A167007 log(A(x)) = 2*x + 6*x^2/2 + 56*x^3/3 + 1810*x^4/4 + 206252*x^5/5 + 86874564*x^6/6 +...+ A167010(n)*x^n/n +...
%o A167007 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^m)*x^m/m)+x*O(x^n)),n)}
%Y A167007 Cf. A167010, A155200, A167006.
%K A167007 nonn,new
%O A167007 0,2
%A A167007 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A167006
%S A167006 1,2,6,66,4258,1337374,1933082159,11353941470188,291885138650054688,
%T A167006 29463501750534915665304,12844314786465829040693498639,
%U A167006 21675661852919288704454219459892060
%N A167006 G.f.: A(x) = exp( Sum_{n>=1} A167009(n)*x^n/n ) where A167009(n) = Sum_{k=0..n} C(n^2,k*n).
%e A167006 G.f.: A(x) = 1 + 2*x + 6*x^2 + 66*x^3 + 4258*x^4 + 1337374*x^5 +...
%e A167006 log(A(x)) = 2*x + 8*x^2/2 + 170*x^3/3 + 16512*x^4/4 + 6643782*x^5/5 + 11582386286*x^6/6 +...+ A167009(n)*x^n/n +...
%o A167006 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m^2,k*m))*x^m/m)+x*O(x^n)),n)}
%Y A167006 Cf. A167009, A155200.
%K A167006 nonn,new
%O A167006 0,2
%A A167006 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A167008
%S A167008 1,2,4,14,106,1732,66634,5745700,1058905642,461715853196,
%T A167008 461918527950694,989913403174541980,5009399946447021173140,
%U A167008 60070720443204091719085184,1548154498059133199618813305334
%N A167008 a(n) = Sum_{k=0..n} C(n,k)^k.
%e A167008 The triangle of coefficients C(n,k)^k, n>=k>=0, begins:
%e A167008 1;
%e A167008 1, 1;
%e A167008 1, 2, 1;
%e A167008 1, 3, 9, 1;
%e A167008 1, 4, 36, 64, 1;
%e A167008 1, 5, 100, 1000, 625, 1;
%e A167008 1, 6, 225, 8000, 50625, 7776, 1;
%e A167008 1, 7, 441, 42875, 1500625, 4084101, 117649, 1;
%e A167008 1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152, 1; ...
%e A167008 in which the row sums form this sequence.
%o A167008 (PARI) a(n)=sum(k=0,n,binomial(n,k)^k)
%Y A167008 Cf. A014062, A000169, A167009, A167010.
%K A167008 nonn,new
%O A167008 0,2
%A A167008 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A167010
%S A167010 1,2,6,56,1810,206252,86874564,132282417920,770670360699138,
%T A167010 16425660314368351892,1367610300690018553312276,
%U A167010 419460465362069257397304825200,509571049488109525160616367158261124
%N A167010 a(n) = Sum_{k=0..n} C(n,k)^n.
%e A167010 The triangle of coefficients C(n,k)^n, n>=k>=0, begins:
%e A167010 1;
%e A167010 1, 1;
%e A167010 1, 4, 1;
%e A167010 1, 27, 27, 1;
%e A167010 1, 256, 1296, 256, 1;
%e A167010 1, 3125, 100000, 100000, 3125, 1;
%e A167010 1, 46656, 11390625, 64000000, 11390625, 46656, 1; ...
%e A167010 in which the row sums form this sequence.
%o A167010 (PARI) a(n)=sum(k=0,n,binomial(n,k)^n)
%Y A167010 Cf. A014062, A000312, A066300, A167009.
%K A167010 nonn,new
%O A167010 0,2
%A A167010 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A167009
%S A167009 1,2,8,170,16512,6643782,11582386286,79450506979090,2334899414608412672,
%T A167009 265166261617029717011822,128442558588779813655233443038,
%U A167009 238431997806538515396060130910954852
%N A167009 a(n) = Sum_{k=0..n} C(n^2,k*n).
%e A167009 The triangle of coefficients C(n^2,k*n), n>=k>=0, begins:
%e A167009 1;
%e A167009 1, 1;
%e A167009 1, 6, 1;
%e A167009 1, 84, 84, 1;
%e A167009 1, 1820, 12870, 1820, 1;
%e A167009 1, 53130, 3268760, 3268760, 53130, 1;
%e A167009 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1; ...
%e A167009 in which the row sums form this sequence.
%o A167009 (PARI) a(n)=sum(k=0,n,binomial(n^2,k*n))
%Y A167009 Cf. A014062, A167010.
%K A167009 nonn,new
%O A167009 0,2
%A A167009 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2009

%I A168062
%S A168062 3,10,24,105,112,120,165,252,680,720,728,858,1320,1365,1771,2280,2907,
%T A168062 3360,3654,3990,4896,5200,6545,6900,8775,9240,9920,10660,12144,13485,
%U A168062 16215,16368,16872,19656,23310,23426,24360,26488,27417,32509,35904
%N A168062 Denominator((n+3)/(n+2)/(n+1)/n) (Sorted with no repeats).
%t A168062 Take[Union@Table[Denominator[(n+3)/(n+2)/(n+1)/n],{n,200}],120]
%Y A168062 Cf. A060789, A168061
%K A168062 nonn,new
%O A168062 1,1
%A A168062 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168061
%S A168061 3,24,10,120,105,112,252,720,165,1320,858,728,1365,3360,680,4896,2907,
%T A168061 2280,3990,9240,1771,12144,6900,5200,8775,19656,3654,24360,13485,9920,
%U A168061 16368,35904,6545,42840,23310,16872,27417,59280,10660,68880,37023,26488
%N A168061 Denominator((n+3)/(n+2)/(n+1)/n).
%C A168061 Numerators((n+3)/(n+2)/(n+1)/n)=A060789.
%t A168061 Table[Denominator[(n+3)/(n+2)/(n+1)/n],{n,60}]
%Y A168061 Cf. A060789
%K A168061 nonn,new
%O A168061 1,1
%A A168061 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168060
%S A168060 2,3,10,12,21,30,36,55,56,78,90,105,132,136,171,182,210,240,253,300,306,
%T A168060 351,380,406,462,465,528,552,595,650,666,741,756,820,870,903,990,992,
%U A168060 1081,1122,1176,1260,1275,1378,1406,1485,1560,1596,1711,1722,1830,1892
%N A168060 Denominator((n+2)/(n+1)/n) (Sorted with no repeats).
%t A168060 Union@Table[Denominator[(n+2)/(n+1)/n],{n,80}]
%Y A168060 Cf. A026741, A168059
%K A168060 nonn,new
%O A168060 1,1
%A A168060 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168059
%S A168059 2,3,12,10,30,21,56,36,90,55,132,78,182,105,240,136,306,171,380,210,462,
%T A168059 253,552,300,650,351,756,406,870,465,992,528,1122,595,1260,666,1406,741,
%U A168059 1560,820,1722,903,1892,990,2070,1081,2256,1176,2450,1275,2652,1378
%N A168059 Denominator((n+2)/(n+1)/n).
%C A168059 Numerators((n+2)/(n+1)/n)=A026741.
%e A168059 3/2/1=3/2, 4/3/2=2/3, 5/4/3=5/12, 6/5/4=3/10, ..
%t A168059 Table[Denominator[(n+2)/(n+1)/n],{n,60}]
%Y A168059 Cf. A026741
%K A168059 nonn,new
%O A168059 1,1
%A A168059 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168058
%S A168058 1,0,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,
%T A168058 83670,227268,621144,1706934,4713558,13072764,36398568,
%U A168058 101704038,285095118,801526446,2259520830,6385455594,18086805002
%V A168058 1,0,-2,-2,-4,-8,-18,-42,-102,-254,-646,-1670,-4376,-11596,-31022,
%W A168058 -83670,-227268,-621144,-1706934,-4713558,-13072764,-36398568,
%X A168058 -101704038,-285095118,-801526446,-2259520830,-6385455594,-18086805002
%N A168058 Expansion of 1-sqrt(1-x-sqrt(1-2x-3x^2)).
%C A168058 a(n+2)=-2*A001006(n). Hankel transform is (-1)^n*A168057(n).
%F A168058 a(n)=0^n-2*sum{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
%Y A168058 Cf. A168055.
%K A168058 easy,sign,new
%O A168058 0,3
%A A168058 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168057
%S A168057 1,2,12,40,80,224,576,1152,2816,6656,13312,30720,69632,139264,311296,
%T A168057 688128,1376256,3014656,6553600,13107200,28311552,60817408,121634816,
%U A168057 260046848,553648128,1107296256,2348810240,4966055936,9932111872
%N A168057 Expansion of (1+8x^2+8x^3)/((1-2x)^2*(1+2x+4x^2)).
%C A168057 Hankel transform of A168058 is (-1)^n*a(n).
%F A168057 a(n)=2^n*A168056(n).
%K A168057 easy,nonn,new
%O A168057 0,2
%A A168057 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168056
%S A168056 1,1,3,5,5,7,9,9,11,13,13,15,17,17,19,21,21,23,25,25,27,29,29,31,33,33,
%T A168056 35,37,37,39,41,41,43,45,45,47,49,49,51,53,53,55,57,57,59,61,61,63,65,
%U A168056 65,67,69,69,71,73,73,75,77,77,79,81,81,83,85,85,87,89,89,91,93,93
%N A168056 Expansion of (1+2x^2+x^3)/((1-x)^2*(1+x+x^2)).
%F A168056 a(n)=A168057(n)/2^n.
%Y A168056 Cf. A168053.
%K A168056 easy,nonn,new
%O A168056 0,3
%A A168056 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168055
%S A168055 1,0,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,83670,227268,
%T A168055 621144,1706934,4713558,13072764,36398568,101704038,285095118,801526446,
%U A168055 2259520830,6385455594,18086805002,51339636952,146015545604
%N A168055 Expansion of 1+sqrt(1-x-sqrt(1-2x-3x^2)).
%C A168055 a(n+2)=2*A001006(n). Hankel transform is A168054.
%F A168055 a(n)=0^n+2*sum{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
%Y A168055 Cf. A168049.
%K A168055 easy,nonn,new
%O A168055 0,3
%A A168055 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168054
%S A168054 1,2,4,24,48,160,448,896,2304,5632,11264,26624,61440,122880,
%T A168054 278528,622592,1245184,2752512,6029312,12058624,26214400,
%U A168054 56623104,113246208,243269632,520093696,1040187392,2214592512
%V A168054 1,2,-4,-24,-48,-160,-448,-896,-2304,-5632,-11264,-26624,-61440,-122880,
%W A168054 -278528,-622592,-1245184,-2752512,-6029312,-12058624,-26214400,
%X A168054 -56623104,-113246208,-243269632,-520093696,-1040187392,-2214592512
%N A168054 Expansion of (1-8x^2-24x^3)/((1-2x)^2*(1+2x+4x^2)).
%C A168054 Hankel transform of A168055.
%F A168054 a(n)=2^n*A168053(n).
%K A168054 easy,sign,new
%O A168054 0,2
%A A168054 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168053
%S A168053 1,1,1,3,3,5,7,7,9,11,11,13,15,15,17,19,19,21,23,23,
%T A168053 25,27,27,29,31,31,33,35,35,37,39,39,41,43,43,45,47,
%U A168053 47,49,51,51,53,55,55,57,59,59,61,63,63,65,67,67,69
%V A168053 1,1,-1,-3,-3,-5,-7,-7,-9,-11,-11,-13,-15,-15,-17,-19,-19,-21,-23,-23,
%W A168053 -25,-27,-27,-29,-31,-31,-33,-35,-35,-37,-39,-39,-41,-43,-43,-45,-47,
%X A168053 -47,-49,-51,-51,-53,-55,-55,-57,-59,-59,-61,-63,-63,-65,-67,-67,-69
%N A168053 Expansion of (1-2x-3x^2)/((1-x)^2*(1+x+x^2)).
%F A168053 a(n)=-(n^9-45n^8+846n^7-8610n^6+51345n^5-181125n^4+361584n^3-361260n^2+137264n-6720)/6720;
%F A168053 a(n)=A168054(n)/2^n.
%K A168053 easy,sign,new
%O A168053 0,4
%A A168053 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168052
%S A168052 1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,11,12,13,13,14,15,15,16,
%T A168052 17,17,18,19,19,20,21,21,22,23,23,24,25,25,26,27,27,28,29,29,
%U A168052 30,31,31,32,33,33,34,35,35,36,37,37,38,39,39,40,41,41,42,43
%V A168052 1,-1,2,-3,3,-4,5,-5,6,-7,7,-8,9,-9,10,-11,11,-12,13,-13,14,-15,15,-16,
%W A168052 17,-17,18,-19,19,-20,21,-21,22,-23,23,-24,25,-25,26,-27,27,-28,29,-29,
%X A168052 30,-31,31,-32,33,-33,34,-35,35,-36,37,-37,38,-39,39,-40,41,-41,42,-43
%N A168052 Hankel transform of a Motzkin variant.
%C A168052 Hankel transform of A168051.
%F A168052 G.f.: (1+x^2)/((1+x)^2(1-x+x^2));
%F A168052 a(n)=cos(pi*n/3)/3+sqrt(3)*sin(pi*n/3)/9+2(n+1)(-1)^n/3.
%K A168052 easy,sign,new
%O A168052 0,3
%A A168052 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168051
%S A168051 1,0,1,1,2,4,9,21,51,127,323,835,2188,5798,15511,41835,
%T A168051 113634,310572,853467,2356779,6536382,18199284,50852019,
%U A168051 142547559,400763223,1129760415,3192727797,9043402501,25669818476
%V A168051 1,0,-1,-1,-2,-4,-9,-21,-51,-127,-323,-835,-2188,-5798,-15511,-41835,
%W A168051 -113634,-310572,-853467,-2356779,-6536382,-18199284,-50852019,
%X A168051 -142547559,-400763223,-1129760415,-3192727797,-9043402501,-25669818476
%N A168051 Expansion of (1+x+sqrt(1-2x-3x^2))/2.
%C A168051 A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.
%Y A168051 Cf. A168049.
%K A168051 easy,sign,new
%O A168051 0,5
%A A168051 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168050
%S A168050 1,1,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,
%T A168050 11,11,12,12,13,13,14,14,15,15,16,16,17,17,18,18,19,
%U A168050 19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,27,27
%V A168050 1,1,0,-1,-1,-2,-2,-3,-3,-4,-4,-5,-5,-6,-6,-7,-7,-8,-8,-9,-9,-10,-10,
%W A168050 -11,-11,-12,-12,-13,-13,-14,-14,-15,-15,-16,-16,-17,-17,-18,-18,-19,
%X A168050 -19,-20,-20,-21,-21,-22,-22,-23,-23,-24,-24,-25,-25,-26,-26,-27,-27
%N A168050 Hankel tranform of Motzkin variant.
%C A168050 Hankel transform of A168049.
%F A168050 G.f.: (1-2x^2-x^3-x^4)/((1+x)(1-x)^2);
%F A168050 a(n)=(-1)^n/4-(2n-3)/4+C(1,n)-C(0,n).
%K A168050 easy,sign,new
%O A168050 0,6
%A A168050 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A168049
%S A168049 1,0,1,1,2,4,9,21,51,127,323,835,2188,5798,15511,41835,113634,310572,
%T A168049 853467,2356779,6536382,18199284,50852019,142547559,400763223,
%U A168049 1129760415,3192727797,9043402501,25669818476,73007772802,208023278209
%N A168049 Expansion of (3-x-sqrt(1-2x-3x^2))/2.
%C A168049 A variant of the Motzkin numbers A001006. Hankel transform is A168050.
%Y A168049 Cf. A168051.
%K A168049 easy,nonn,new
%O A168049 0,5
%A A168049 Paul Barry (pbarry(AT)wit.ie), Nov 17 2009

%I A167928
%S A167928 1,0,0,0,0,1,1,3,4,6,9,13,16
%N A167928 Number of partitions of n that do not contain 1 as a part and whose parts are not the same divisor of n.
%C A167928 Note that these partitions are located in the head of the outer shell of the partitions of n (See the shell model of partitions, here).
%H A167928 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>
%H A167928 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A167928 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A167928 a(n) = A002865(n) - A032741(n).
%e A167928 The partitions of 6 are:
%e A167928 6 ....................... All parts are the same divisor of n.
%e A167928 5 + 1 ................... Contains 1 as a part.
%e A167928 4 + 2 ................... All parts are not the same divisor of n. <------(1)
%e A167928 4 + 1 + 1 ............... Contains 1 as a part.
%e A167928 3 + 3 ................... All parts are the same divisor of n.
%e A167928 3 + 2 + 1 ............... Contains 1 as a part.
%e A167928 3 + 1 + 1 + 1 ........... Contains 1 as a part.
%e A167928 2 + 2 + 2 ............... All parts are the same divisor of n.
%e A167928 2 + 2 + 1 + 1 ........... Contains 1 as a part.
%e A167928 2 + 1 + 1 + 1 + 1 ....... Contains 1 as a part.
%e A167928 1 + 1 + 1 + 1 + 1 + 1 ... Contains 1 as a part.
%e A167928 Then a(6) = 1.
%Y A167928 Cf. A000041, A002865, A032741, A144300, A135010, A138121, A167929, A167930, A167932, A167934.
%K A167928 more,nonn,new
%O A167928 0,8
%A A167928 Omar E. Pol (info(AT)polprimos.com), Nov 17 2009

%I A168048
%S A168048 1,2,4,6,9,12,16,20,24,30,35,42,48,54,60
%N A168048 C(n)*Pi(n), C(n) = number of nonprimes <=n, Pi(n)= number of primes <=n; n = 2,3,4,5,...A000720*A062298
%Y A168048 Cf. A000720, A062298
%K A168048 nonn,new
%O A168048 1,2
%A A168048 Daniel Tisdale (daniel6874(AT)gmail.com), Nov 17 2009

%I A168047
%S A168047 1,2,3,4,12,23,70
%N A168047 n-th single or isolated number divides n.
%Y A168047 Cf. A000027, A167706.
%K A168047 nonn,new
%O A168047 1,2
%A A168047 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 17 2009

%I A168045
%S A168045 1,3,6,9,12,14,16,18,20,23,25,27,29,31,34,36,38,40,43,45,47,49,51,53,56,
%T A168045 58,60,62,64,66,69,71,73,75,78,80,82,84,87,89,91,93,95,98,100,102,104,
%U A168045 106,108,111,113,115,117,119,121,124,126,128,130,133,135,137,139,141
%N A168045 n-th non-single or nonisolated number plus n.
%F A168045 a(n)=A167707(n)+A000027(n).
%Y A168045 Cf. A000027, A167707.
%K A168045 nonn,new
%O A168045 1,2
%A A168045 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 17 2009

%I A168044
%S A168044 0,4,5,7,8,10,11,12,13,14,16,17,18,19,20,22,23,24,25,26,27,28,29,31,32,
%T A168044 33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,50,52,53,55,56,57,58,
%U A168044 59,60,61,62,63,64,65,66,67,68,70,71,72,73,74,76,77,78,79,80,81,82,83
%N A168044 Bisection of the even nonisolated nonprimes A167692.
%F A168044 a(n)=A167692(n)/2.
%Y A168044 Cf. A167692.
%K A168044 nonn,new
%O A168044 1,2
%A A168044 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 17 2009

%I A168042
%S A168042 0,1,9,10,11,14,19,40,41,44,49,90,91,94,99,100,101,104,109,110,111,114,
%T A168042 119,140,141,144,149,190,191,194,199,400,404,410,411,414,419,440,441,
%U A168042 444,490,494,900,901,904,909,910,914,940,944,949,990,994,999,1000,1001
%N A168042 The non-single or nonisolated numbers whose digits are all square.
%C A168042 Square digits are 0, 1, 4 or 9 (i.e. 0=0*0, 1=1*1, 4=2*2, 9=3*3). The non-single or nonisolated numbers are 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21,..
%Y A168042 Cf. A000290, A157908, A167707.
%K A168042 nonn,new
%O A168042 1,3
%A A168042 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 17 2009

%I A168043
%S A168043 1,2,4,7,13,23,40,68,114,189,311,509,830,1350,2192,3555
%N A168043 Let S(1)={1} and, for n>1 let S(n) be the smallest set containing x+1, x+2, and 2*x for each element x in S(n-1). a(n) is the number of elements in S(n).
%F A168043 It appears that a(n)=a(n-1)+a(n-2)+n-3, for n>3.
%e A168043 Under the indicated set mapping we have {1} -> {2,3} -> {3,4,5,6} -> {4,5,6,7,8,10,12},..., so a(2)=2, a(3)=4, a(4)=7, etc.
%Y A168043 Cf. A122554.
%K A168043 nonn,new
%O A168043 1,2
%A A168043 John W. Layman (layman(AT)math.vt.edu), Nov 17 2009

%I A168041
%S A168041 1,5,144,46368,75025,14930352,4807526976,1548008755920,498454011879264,
%T A168041 51680708854858323072,16641027750620563662096,5358359254990966640871840,
%U A168041 59425114757512643212875125,555565404224292694404015791808
%N A168041 Fibonacci numbers if Fibonacci(n)/n are Integers.
%t A168041 f[n_]:=Fibonacci[n]/n; lst={};Do[If[IntegerQ[f[n]],AppendTo[lst,Fibonacci[n]]],{n,4*5!}];lst
%Y A168041 Cf. A167745
%K A168041 nonn,new
%O A168041 1,2
%A A168041 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168040
%S A168040 3,5,11,97,389,463,877,937,1049,1061,1283,1307,1319,1453,1579,1657,1907,
%T A168040 2081,2143,2339,2351,2383,2459,2687,2741,3061,3433,3547,3581,4027,4241,
%U A168040 4363,4447,4481,4831,4903,4919,4973,5171,5737,5939,6257,6269,6299,7159
%N A168040 Primes p for which floor(p^E) is prime.
%t A168040 $MaxExtraPrecision=7!; Select[Prime[Range[3*6! ]],PrimeQ[Floor[ #^E]]&]
%K A168040 nonn,new
%O A168040 1,1
%A A168040 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168039
%S A168039 0,4,4,9,9,16,16,25,25,25,25,36,36,36,36,49,49,49,49,64,64,64,64,64,64,
%T A168039 81,81,81,81,81,81,100,100,100,100,100,100,121,121,121,121,121,121,121,
%U A168039 121,144,144,144,144,144,144,144,144,169,169,169,169,169,169,169,169
%N A168039 Squares closest to 3*n.
%t A168039 Round[Sqrt[3*Range[0,120]]]^2
%Y A168039 Cf. A168038.
%K A168039 nonn,new
%O A168039 0,2
%A A168039 Zak Seidov (zakseidov(AT)yahoo.com), Nov 17 2009

%I A168038
%S A168038 0,1,4,4,9,9,9,16,16,16,16,25,25,25,25,25,36,36,36,36,36,36,49,49,49,49,
%T A168038 49,49,49,64,64,64,64,64,64,64,64,81,81,81,81,81,81,81,81,81,100,100,
%U A168038 100,100,100,100,100,100,100,100,121,121,121,121,121,121,121,121,121
%N A168038 Squares closest to 2*n.
%t A168038 Round[Sqrt[2*Range[0,120]]]^2
%Y A168038 Cf. A093995.
%K A168038 nonn,new
%O A168038 0,3
%A A168038 Zak Seidov (zakseidov(AT)yahoo.com), Nov 17 2009

%I A168037
%S A168037 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1,0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,
%T A168037 1,0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1
%N A168037 0,1,2,9,8,25,18,49,32,=A129194 mod 9.
%C A168037 See A154811=period 12: 1,2,5,4,7,8,8,7,4,5,2,1.Palindrom.
%F A168037 Period 18:repeat 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1. Palindrom for 19 and 17 terms.
%K A168037 nonn,uned,new
%O A168037 0,3
%A A168037 Paul Curtz (bpcrtz(AT)free.fr), Nov 17 2009

%I A168036
%S A168036 0,1,1,2,0,4,1,6,4,3,3,10,4,12,5,7,16,16,3,18,4,11,9,
%T A168036 22,20,15,11,0,4,28,1,30,48,19,15,23,24,36,17,23,28,40,1,
%U A168036 42,4,6,21,46,64,35,5,31,4,52,27,39,36,35,27,58,32,60,29
%V A168036 0,-1,-1,-2,0,-4,-1,-6,4,-3,-3,-10,4,-12,-5,-7,16,-16,3,-18,4,-11,-9,
%W A168036 -22,20,-15,-11,0,4,-28,1,-30,48,-19,-15,-23,24,-36,-17,-23,28,-40,-1,
%X A168036 -42,4,-6,-21,-46,64,-35,-5,-31,4,-52,27,-39,36,-35,-27,-58,32,-60,-29
%N A168036 Difference between n' and n, being n' the arithmetic derivative of n.
%C A168036 Let k=n'-n. For k=-1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k=1 n is a Giuga number (A007850).
%p A168036 P:= proc(p) local a,b,m,n,i,ok,t1,t2,t3; a:=0; for n from 0 by 1 to p do b:=1000000000039; ok:=0; if n<=1 then a:=0; ok:=1; fi; if isprime(n) then a:=1; ok:=1; fi; if ok=0 then t1:=ifactor(b*n); m:=nops(t1); t2:=0; for i from 1 to m do t3:=op(i,t1); if nops(t3)=1 then t2:=t2+1/op(t3); else t2:=t2+op(2,t3)/op(op(1,t3)); fi; od; t2:=t2-1/b; a:=n*t2; fi; print(a-n); od; end: P(1000);
%Y A168036 Cf. A007850, A051674, A054377
%K A168036 easy,sign,uned,new
%O A168036 0,4
%A A168036 Paolo P. Lava (ppl(AT)spl.at), Nov 17 2009

%I A168035
%S A168035 2,5,7,17,61,617,7741,10691
%N A168035 Primes p for which floor(p^GoldenRatio) and floor(GoldenRatio^p) are also primes.
%t A168035 $MaxExtraPrecision=8!; Select[Prime[Range[3*6! ]],PrimeQ[Floor[ #^GoldenRatio]]&&PrimeQ[Floor[GoldenRatio^# ]]&]
%K A168035 nonn,new
%O A168035 1,1
%A A168035 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168034
%S A168034 2,3,5,7,17,43,59,61,109,229,263,269,419,443,457,617,997,1069,1301,1373,
%T A168034 1483,1523,1543,1801,1877,1949,2053,2269,2309,2411,2503,2551,2633,2731,
%U A168034 2741,2887,2963,3023,3181,3323,3359,3571,3607,3673,4129,4153,4423,4483
%N A168034 Primes p for which floor(p^GoldenRatio) is prime.
%t A168034 $MaxExtraPrecision=6!; Select[Prime[Range[5! ]],PrimeQ[Floor[ #^GoldenRatio]]&]
%K A168034 nonn,new
%O A168034 1,1
%A A168034 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168033
%S A168033 2,5,7,11,13,17,19,31,37,41,47,53,61,71,79,113,313,353,503,613,617
%N A168033 Primes p for which floor(GoldenRatio^p) is prime.
%t A168033 $MaxExtraPrecision=6!; Select[Prime[Range[5! ]],PrimeQ[Floor[GoldenRatio^# ]]&]
%K A168033 nonn,new
%O A168033 1,1
%A A168033 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168032
%S A168032 2,7,11,23,37,79,101,107,131,149,163,227,241,283,311,353,367,379,383,
%T A168032 409,419,457,487,509,613,661,719,761,797,971,997,1049,1279,1321,1373,
%U A168032 1447,1451,1453,1483,1531,1613,1699,1861,1877,2011,2039,2069,2137,2143
%N A168032 Primes p for which floor(GoldenRatio*p) is prime.
%t A168032 Select[Prime[Range[6! ]],PrimeQ[Floor[GoldenRatio*# ]]&]
%K A168032 nonn,new
%O A168032 1,1
%A A168032 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168031
%S A168031 2,5,7,11,47,89,97,113,163,227,283,367,373,431,439,503,643,823,877,887,
%T A168031 941,991,1013,1049,1093,1303,1327,1439,1523,1567,1609,1879,1901,1949,
%U A168031 1951,1993,2113,2179,2221,2347,2399,2411,2477,2503,2591,2689,2711,2797
%N A168031 Primes p for which floor(E*p) is prime.
%t A168031 Select[Prime[Range[6! ]],PrimeQ[Floor[E*# ]]&]
%K A168031 nonn,new
%O A168031 1,1
%A A168031 Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009

%I A168030
%S A168030 1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,
%T A168030 0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,
%U A168030 1,0,0,0,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,1,1,0
%N A168030 Variant of pendular triangle A118340.
%C A168030 Replaced the sums (f(a,b)=a+b) by the operators f(a,b)=a^2-ab+b^2 in the construction of triangle in A118340.
%K A168030 nonn,tabl,new
%O A168030 0,1
%A A168030 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2009

%I A168028
%S A168028 4903,4567,4243,3931,3631,3343,3067,2803,2551,2311,2083,1867,1663,1471,
%T A168028 1291,1123,967,823,691,571,463,367,283,211,151,103,67,43,31,31,43,67,
%U A168028 103,151,211,283,367,463,571,691,823,967,1123,1291,1471,1663,1867,2083
%N A168028 Consecutive non-negative primes of the form 6n^2-342n+4903.
%F A168028 a(n)=6*n^2-342*n+4903
%e A168028 a(0)=4903 a(1)=4567 a(2)=4243 ... a(58)=4903
%K A168028 nonn,new
%O A168028 1,1
%A A168028 Bobby Kramer & Adam Avello (panthar1(AT)gmail.com), Nov 16 2009

%I A168012
%S A168012 8,48,133,302
%N A168012 a(n) = Sum of all divisors of all numbers k such that n^2=< k <(n+1)^2.
%e A168012 For n=2 the a(2)=48 because the numbers k are 4,5,6,7 and 8. (Because 2^2<= k <3^2). Then a(2)= sigma(4)+sigma(5)+sigma(6)+sigma(7)+sigma(8) = 7+6+12+8+15 = 48, where sigma(n) is the sum of divisor of n (See A000203).
%Y A168012 Cf. A000203, A024916, A168010, A168011, A168013.
%K A168012 more,nonn,new
%O A168012 1,1
%A A168012 Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A168013
%S A168013 8,56,189,491
%N A168013 a(n) = Sum of all divisors of all numbers < (n+1)^2.
%C A168013 Partial sums of A168012.
%e A168013 For n=2 the a(2)=56 because the numbers < (2+1)^2 are 1,2,3,4,5,6,7 and 8. Then a(2)= sigma(1))+sigma(2)+sigma(3)+sigma(4)+sigma(5)+sigma(6)+sigma(7)+sigma(8) = 1+3+4+7+6+12+8+15 = 56, where sigma(n) is the sum of divisor of n (See A000203).
%Y A168013 Cf. A000203, A024916, A168010, A168011, A168012.
%K A168013 more,nonn,new
%O A168013 1,1
%A A168013 Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A168011
%S A168011 5,20,45,84
%N A168011 a(n) = Sum of all numbers of divisors of all numbers < (n+1)^2.
%C A168011 Partial sums of A168010.
%e A168011 For n=2 the a(2)=20 because the numbers < (2+1)^2 are 1,2,3,4,5,6,7 and 8. Then a(2)= d(1)+d(2)+d(3)+d(4)+d(5)+d(6)+d(7)+d(8) = 1+2+2+3+2+4+2+4 = 20, where d(n) is the number of divisor of n (See A000005).
%Y A168011 Cf. A000005, A168010, A168012, A168013.
%K A168011 more,nonn,new
%O A168011 1,1
%A A168011 Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A168010
%S A168010 5,15,25,39
%N A168010 a(n) = Sum of all numbers of divisors of all numbers k such that n^2=< k <(n+1)^2.
%e A168010 For n=2 the a(2)=15 because the numbers k are 4,5,6,7 and 8. (Because 2^2<= k <3^2). Then a(2)= d(4)+d(5)+d(6)+d(7)+d(8) = 3+2+4+2+4 = 15, where d(n) is the number of divisor of n (See A000005).
%Y A168010 Cf. A000005, A168011, A168012, A168013.
%K A168010 more,nonn,new
%O A168010 1,1
%A A168010 Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A168027
%S A168027 1,23,163,281,431,613,827,2003,2377,3221,3691,6521,7877,10151,10973,
%T A168027 11827,12713,17623,18701,23333,24571,25841,27143,28477,38711,43577,
%U A168027 45263,48731,50513,65921,72227,81083,85703,95327,97813,102881,124433
%N A168027 Non-composite numbers in the southern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%F A168027 Positive numbers of the form 4n^2 + 3n + 1 with no more than two divisors.
%t A168027 Select[Table[4 n^2 + 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]
%Y A168027 Cf. A033951, all numbers of the form 4n^2 + 3n + 1. Non-composites of eastern ray are in A168022. Primes of northeastern ray are in A073337. Non-composites of northern ray are in A168023. Primes of northwestern ray are in A121326. Non-composites of western ray are in A168025. Non-composites of southwestern ray are in A168026. There are no primes on the southeastern ray, which, being A016754, are the odd squares, and thus none of them prime.
%K A168027 easy,nonn,new
%O A168027 0,2
%A A168027 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A168026
%S A168026 1,7,43,73,157,211,421,601,1483,2551,2971,3907,4423,6163,6481,8191,
%T A168026 12211,19183,22651,26407,27061,28393,31153,35533,37057,37831,42643,
%U A168026 47743,55933,60763,71023,74257,77563,83233,84391,98911,110557,113233
%N A168026 Non-composite numbers in the southwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%F A168026 Positive numbers of the form 4n^2 - 6n + 3 with no more than two divisors.
%t A168026 Select[Table[4 n^2 - 6 n + 3, {n, 200}], Length[Divisors[ # ]] < 3 &]
%Y A168026 Cf. A054569, all numbers of the form 4n^2 - 6n + 3. Non-composites of eastern ray are in A168022. Primes of northeastern ray are in A073337. Non-composites of northern ray are in A168023. Primes of northwestern ray are in A121326. Non-composites of western ray are in A168025. Non-composites of southern ray are in A168027.
%K A168026 easy,nonn,new
%O A168026 1,2
%A A168026 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A168025
%S A168025 1,19,151,1621,2731,3631,4129,7789,11719,12601,14461,15439,17491,20809,
%T A168025 28309,29671,32491,41719,59659,69829,78541,83089,85411,92569,97501,
%U A168025 115771,132679,138571
%N A168025 Non-composite numbers in the western ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%F A168025 Positive numbers of the form 4n^2 - 7n + 4 with no more than two divisors.
%t A168025 Select[Table[4 n^2 - 7 n + 4, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]
%Y A168025 Cf. A054567.
%K A168025 easy,nonn,new
%O A168025 0,2
%A A168025 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A168024
%S A168024 1,5,17,37,101,197,257,401,577,677,1297,1601,2917,3137,4357,5477,7057,
%T A168024 8101,8837,12101,13457,14401,15377,15877,16901,17957,21317,22501,24337,
%U A168024 25601,28901,30977,32401,33857
%N A168024 Non-composite numbers in the northwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%F A168024 Positive numbers of the form 4n^2 + 1 with no more than two divisors.
%t A168024 Select[Table[4 n^2 + 1, {n, 0, 99}], Length[Divisors[ # ]] < 3 &]
%Y A168024 Essentially the same sequence as A002496, A121326, A163588.
%Y A168024 Cf. A053755, all numbers of the form 4n^2 + 1. Non-composites of eastern ray are in A168022. Primes of northeastern ray are in A073337. Non-composites of northern ray are in A168023. Primes of northwestern ray are in A121326 (the same as this sequence but without the initial 1). Non-composites of western ray are in A168025. Non-composites of southwestern ray are in A168026. Non-composites of southern ray are in A168027.
%K A168024 easy,nonn,new
%O A168024 0,2
%A A168024 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A168023
%S A168023 1,61,139,1009,1279,2281,3109,3571,4591,6361,8419,13399,14341,17359,
%T A168023 19531,23029,35251,39901,44839,46549,51871,55579,61381,73849,76039,
%U A168023 102241,110059,135241,153469,156619
%N A168023 Non-composite numbers in the northern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%F A168023 Positive numbers of the form 4n^2 - 9n + 6 with no more than two divisors.
%t A168023 Select[Table[4 n^2 - 9 n + 6, {n, 200}], Length[Divisors[ # ]] < 3 &]
%Y A168023 Cf. A054556, all numbers of the form 4n^2 - 9n + 6. Non-composites of eastern ray are in A168022. Primes of northeastern ray are in A073337. Non-composites of northwestern ray are in A168024. Non-composites of western ray are in A168025. Non-composites of southwestern ray are in A168026. Non-composites of southern ray are in A168027.
%K A168023 easy,nonn,new
%O A168023 1,2
%A A168023 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A168022
%S A168022 1,2,11,53,127,233,541,743,977,1871,3511,4001,4523,5077,9851,11503,
%T A168022 12377,14221,16193,19391,20521,21683,22877,24103,29327,30713,33581,
%U A168022 42953,55343,57241,63127,67211,80231,84827,91961,101921,104491,123377
%N A168022 Non-composite numbers in the eastern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.
%C A168022 Although 1 was not considered a prime number in Ulam's time, the March 1964 cover of Scientific American shows 1 highlighted in the same way as the primes.
%C A168022 "East" and "West" as given here match the Scientific American cover, however, ""North" and "South" are swi
%H A168022 YouTube, <a href="http://www.youtube.com/watch?v=dx24qqBc-PY">Ulam spiral</a>. Note that "East" and "West" in this video match the cover of Scientific American, but "North" and "South" are switched.
%H A168022 BackIssues.com, <a href="http://backissues.com/issue/Scientific-American-March-1964">Scientific American March 1964 back issue</a>
%H A168022 Mathworld, <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral</a>
%F A168022 Positive numbers of the form 4n^2 - 3n + 1 with no more than two divisors.
%t A168022 Select[Table[4 n^2 - 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]
%Y A168022 Cf. A054552, all numbers of the form 4n^2 - 3n + 1. Primes of northeastern ray are in A073337. Non-composites of northern ray are in A168023. Non-composites of northwestern ray are in A168024. Non-composites of western ray are in A168025. Non-composites of southwestern ray are in A168026. Non-composites of southern ray are in A168027.
%K A168022 easy,nonn,new
%O A168022 0,2
%A A168022 Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 16 2009

%I A167934
%S A167934 1,1,1,2,3,6,8,14,19,28,39,55,72,100,132,173,227
%N A167934 a(n) = A000041(n) - A032741(n).
%C A167934 a(n) is also the number of partitions of n whose parts are not equal (if n is included).
%C A167934 Note that the number of partitions of n whose parts are equal is equal to the number of divisors of n, for n>0. (See also A144300).
%H A167934 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>
%H A167934 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A167934 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A167934 a(n) = A000041(n) - A032741(n).
%e A167934 The partitions of 6 are:
%e A167934 6 ....................... Included ....................... (1).
%e A167934 5 + 1 ................... All parts are not equal ........ (2).
%e A167934 4 + 2 ................... All parts are not equal ........ (3).
%e A167934 4 + 1 + 1 ............... All parts are not equal ........ (4).
%e A167934 3 + 3 ................... All parts are equal.
%e A167934 3 + 2 + 1 ............... All parts are not equal ........ (5).
%e A167934 3 + 1 + 1 + 1 ........... All parts are not equal ........ (6).
%e A167934 2 + 2 + 2 ............... All parts are equal.
%e A167934 2 + 2 + 1 + 1 ........... All parts are not equal ........ (7).
%e A167934 2 + 1 + 1 + 1 + 1 ....... All parts are not equal ........ (8).
%e A167934 1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
%e A167934 Then a(6) = 8.
%Y A167934 Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.
%K A167934 more,nonn,new
%O A167934 0,4
%A A167934 Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A168006
%S A168006 37,61,71,163,173,193,367,397,479,499,521,541,571,601,631,641,661,691,
%T A168006 701,751,761,811,821,881,911,941,971,991,1523,1543,1553,1583,1613,1663,
%U A168006 1693,1723,1733,1753,1783,1823,1873,1913,1933,1973,1993,3517,3527,3547
%N A168006 Primes n with property that first digit of 2*n = last digit of n.
%C A168006 Or, primes in A167994.
%e A168006 2*37=74, 2*61=122.
%Y A168006 Cf. A167994.
%K A168006 base,nonn,new
%O A168006 1,1
%A A168006 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A168005
%S A168005 21,32,43,54,65,76,87,98,109,121,131,141,151,161,171,181,191,201,211,
%T A168005 221,232,242,252,262,272,282,292,302,312,322,332,343,353,363,373,383,
%U A168005 393,403,413,423,433,443,454,464,474,484,494,504,514,524,534,544,554
%N A168005 Numbers n with property that first digit of 9*n = last digit of n.
%e A168005 9*21=189, 9*32=288, 9*43=387, 9*54=486, etc.
%t A168005 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[9*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A168005 Cf. A167994, A167996, A167997, A167998, A168000, A168001, A168004.
%K A168005 base,nonn,new
%O A168005 1,1
%A A168005 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A168003
%S A168003 7,37,67,97,127,157,255,277,307,337,367,397,457,487,547,577,607,727,757,
%T A168003 787,877,907,915,937,967,997,1087,1117,1237,1245,1297,1327,1447,1567,
%U A168003 1597,1627,1657,1747,1777,1867,1905,1987,2017,2125,2137,2235,2287,2347
%N A168003 Orderly numbers (mod tau(n)+3).
%C A168003 See A167408 for information about orderly numbers. It appears that when n is in this sequence, then tau(n)+3 must be a prime with primitive root 2 (A001122). For each one of those primes, it is possible to find all forms of n that are orderly. In particular, the form n=p^k*q is in this sequence when 2+5k is in A001122. In that case, we have the congruences p=2+tau(n)/2 and q=1+tau(n)/2 (mod tau(n)+3). When tau(n) is a multiple of 8, then another pair of congruences is p=1+tau(n)/2 and q=2+tau(n)/2 (mod tau(n)+3).
%F A168003 An exhaustive search over forms of n having a prime value of tau(n)+3 finds that terms of this sequence satisfy the following congruences for tau(n)+3 < 60.
%F A168003 . p with prime p = 2 mod 5
%F A168003 . p^3*q with primes {p,q} == {5,6} mod 11
%F A168003 . p^3*q with primes {p,q} == {6,5} mod 11
%F A168003 . p*q*r with primes {p,q,r} == {3,5,6} mod 11
%F A168003 . p^4*q with primes {p,q} == {7,6} mod 13
%F A168003 . p^7*q with primes {p,q} == {9,10} mod 19
%F A168003 . p^7*q with primes {p,q} == {10,9} mod 19
%F A168003 . p^3*q*r with primes {p,q,r} == {5,9,10} mod 19
%F A168003 . p^3*q*r with primes {p,q,r} == {9,6,10} mod 19
%F A168003 . p^3*q*r with primes {p,q,r} == {10,6,9} mod 19
%F A168003 . p*q*r*s with primes {p,q,r,s} == {5,6,9,10} mod 19
%F A168003 . p^12*q with primes {p,q} == {15,14} mod 29
%F A168003 . p^16*q with primes {p,q} == {19,18} mod 37
%F A168003 . p^24*q with primes {p,q} == {27,26} mod 53
%F A168003 . p^4*q^4*r with primes {p,q,r} == {5,27,26} mod 53
%F A168003 . p^27*q with primes {p,q} == {29,30} mod 59
%F A168003 . p^27*q with primes {p,q} == {30,29} mod 59
%F A168003 . p^13*q*r with primes {p,q,r} == {15,29,30} mod 59
%F A168003 . p^13*q*r with primes {p,q,r} == {29,30,36} mod 59
%F A168003 . p^13*q*r with primes {p,q,r} == {30,29,36} mod 59
%F A168003 . p^6*q^3*r with primes {p,q,r} == {29,53,30} mod 59
%F A168003 . p^6*q^3*r with primes {p,q,r} == {30,6,29} mod 59
%F A168003 . p^6*q^3*r with primes {p,q,r} == {48,29,30} mod 59
%F A168003 . p^6*q^3*r with primes {p,q,r} == {48,30,29} mod 59
%F A168003 . p^6*q*r*s with primes {p,q,r,s} == {7,28,30,45} mod 59
%F A168003 . p^6*q*r*s with primes {p,q,r,s} == {15,29,30,36} mod 59
%F A168003 . p^6*q*r*s with primes {p,q,r,s} == {29,30,36,53} mod 59
%F A168003 . p^6*q*r*s with primes {p,q,r,s} == {30,6,29,36} mod 59
%F A168003 . p^6*q*r*s with primes {p,q,r,s} == {48,15,29,30} mod 59
%F A168003 Andrew Weimholt found some of these forms.
%K A168003 nonn,new
%O A168003 1,1
%A A168003 T. D. Noe (noe(AT)sspectra.com), Nov 16 2009

%I A168004
%S A168004 21,32,43,54,65,76,86,97,108,119,131,141,151,161,171,181,191,201,211,
%T A168004 221,231,241,252,262,272,282,292,302,312,322,332,342,352,362,372,383,
%U A168004 393,403,413,423,433,443,453,463,473,483,493,504,514,524,534,544,554
%N A168004 Numbers n with property that first digit of 8*n = last digit of n.
%e A168004 8*21=168, 8*32=256, 8*43=344, 8*54=432, etc.
%t A168004 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[8*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A168004 Cf. A167994, A167996, A167997, A167998, A168000, A168001.
%K A168004 base,nonn,new
%O A168004 1,1
%A A168004 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A168001
%S A168001 21,32,42,43,53,64,75,85,86,96,107,118,128,129,139,151,161,171,181,191,
%T A168001 201,211,221,231,241,251,261,271,281,292,302,312,322,332,342,352,362,
%U A168001 372,382,392,402,412,422,433,443,453,463,473,483,493,503,513,523,533
%N A168001 Numbers n with property that first digit of 7*n = last digit of n.
%e A168001 7*21=147, 7*32=224, 7*42=294, 7*43=301, etc.
%t A168001 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[7*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A168001 Cf. A167994, A167996, A167997, A167998, A168000.
%K A168001 base,nonn,new
%O A168001 1,1
%A A168001 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A168000
%S A168000 21,31,42,53,63,74,85,95,106,116,117,127,138,148,159,171,181,191,201,
%T A168000 211,221,231,241,251,261,271,281,291,301,311,321,331,342,352,362,372,
%U A168000 382,392,402,412,422,432,442,452,462,472,482,492,503,513,523,533,543
%N A168000 Numbers n with property that first digit of 6*n = last digit of n.
%e A168000 6*21=126, 6*31=186, 6*42=252, 6*53=318, etc.
%t A168000 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[6*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A168000 Cf. A167994, A167996, A167997, A167998.
%K A168000 base,nonn,new
%O A168000 1,1
%A A168000 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A167999
%S A167999 1,3,10,46,264,1773,13719,120770
%N A167999 A permutation $\pi$ on $\{1,2,....n\}$ has $k(\pi)$ longest increasing subsequences associated with it; $1\le k(\pi)\le f(n)$ for some function $f$. The given sequence enumerates $\sum_{\pi} k(\pi)$.
%C A167999 We also have data for the number of permutations $\pi$ that have $k(\pi)=r$ for $r\ge 1$.
%K A167999 nonn,new
%O A167999 1,2
%A A167999 Anant Godbole, Stephanie Goins, Brad Wild (godbolea(AT)etsu.edu), Nov 16 2009

%I A167998
%S A167998 19,21,31,42,52,63,73,84,94,105,115,126,136,147,157,168,178,189,199,201,
%T A167998 211,221,231,241,251,261,271,281,291,301,311,321,331,341,351,361,371,
%U A167998 381,391,402,412,422,432,442,452,462,472,482,492,502,512,522,532,542
%N A167998 Numbers n with property that first digit of 5*n = last digit of n.
%e A167998 5*19=95, 5*21=105, 5*31=155, 5*42=210, etc.
%t A167998 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[5*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A167998 Cf. A167994, A167996, A167997.
%K A167998 base,nonn,new
%O A167998 1,1
%A A167998 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A167997
%S A167997 16,31,41,52,62,72,83,93,104,114,124,125,135,145,156,166,177,187,197,
%T A167997 208,218,229,239,249,251,261,271,281,291,301,311,321,331,341,351,361,
%U A167997 371,381,391,401,411,421,431,441,451,461,471,481,491,502,512,522,532
%N A167997 Numbers n with property that first digit of 4*n = last digit of n.
%e A167997 4*16=64, 4*31=124, 4*41=164, 4*52=208, etc.
%t A167997 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[4*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A167997 Cf. A167994, A167996.
%K A167997 base,nonn,new
%O A167997 1,1
%A A167997 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A167996
%S A167996 13,14,28,41,51,61,72,82,92,103,113,123,133,134,144,154,164,175,185,195,
%T A167996 206,216,226,237,247,257,268,278,288,298,309,319,329,341,351,361,371,
%U A167996 381,391,401,411,421,431,441,451,461,471,481,491,501,511,521,531,541
%N A167996 Numbers n with property that first digit of 3*n = last digit of n.
%e A167996 3*13=39, 3*14=42, 3*28=84, 3*41=123, etc.
%t A167996 Reap[Do[If[IntegerDigits[n][[ -1]]==IntegerDigits[3*n][[1]],Sow[n]],{n,1000}]][[2,1]]
%Y A167996 Cf. A167994.
%K A167996 base,nonn,new
%O A167996 1,1
%A A167996 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A167995
%S A167995 1,1,3,10,44,238,1506,10960
%N A167995 For n>=1, this sequence gives the total number of permutations on {1,2,...,n} that have a unique longest increasing subsequence
%C A167995 Example: For n=3, 123, 231, and 312 are the only three permutations that have precisely one maximal increasing subsequence.
%K A167995 nonn,nice,new
%O A167995 1,3
%A A167995 Anant Godbole, Stephanie Goins, Brad Wild (godbolea(AT)etsu.edu), Nov 16 2009

%I A167994
%S A167994 12,24,25,37,49,51,61,71,81,91,102,112,122,132,142,153,163,173,183,193,
%T A167994 204,214,224,234,244,255,265,275,285,295,306,316,326,336,346,357,367,
%U A167994 377,387,397,408,418,428,438,448,459,469,479,489,499,501,511,521,531
%N A167994 Numbers n with property that first digit of 2*n = last digit of n.
%e A167994 2*12=24, 2*24=48, 2*25=50, 2*37=74, etc.
%K A167994 base,nonn,new
%O A167994 1,1
%A A167994 Zak Seidov (zakseidov(AT)yahoo.com), Nov 16 2009

%I A167993
%S A167993 0,0,1,3,12,36,117,351,1080,3240
%N A167993 a(n)=3a(n-1)+3a(n-2)-9a(n-3); a(0)=a(1)=0,a(2)=1. Principal sequence (see A138587).
%C A167993 From a(n)=p*a(n-1)+q*a(n-2)-p*q*a(n-3). Here p=q=3. Third of a family with p=q=A000027: 1) a(n)=a(n-1)+a(n-2)-a(n-3) which principal sequence is 0,0,1,1,2,2,3,3,=A004526; 2) a(n)=2a(n-1)+2a(n-2)-4a(n-3) leads to 0,0,1,2,6,12,28,56,=0,A141447=0,0,A122746 which differences A007179=0,1,1,4,6,16,=A156232/4 (and successive) have same recurrence.See A135094,A010036,A006516; 3) a(2n+1)=3*a(2n); 4) a(n)=4a(n-1)+4a(n-4)-16a(n-3).
%F A167993 First differences are 0,1,2,9,24,81,=A122006 (see A108411 in A167936;A122006(2n)=2*a(2n)) . a(n)=3a(n=1) + (0,1,0,3,0,9,0,27,=mix A000004,A000244).
%K A167993 nonn,uned,new
%O A167993 0,4
%A A167993 Paul Curtz (bpcrtz(AT)free.fr), Nov 16 2009

%I A167992
%S A167992 0,0,157,1097,10039
%N A167992 Least n-digit emirp (A006567) with emirp digital sum, or 0 if no such value.
%C A167992 Least emirp (nonpalindromatic prime in A007500, i.e. prime whose reversal is a different prime) greater than 10^n, for which the sum of digits (A007953) is also an emirp.
%F A167992 a(n) = Min{p > 10^n in A006567, and A007953(p) is in in A006567} = Min{p > 10^n in A000040 such that A004086(p) is in A000040, and A004086(p) distinct from p, and in A006567(p) is in A000040, and A004086(p) distinct from A006567(p), and in A000040}.
%e A167992 a(1) = a(2) = 0. a(3) = 157 because 157 is the least nonpalindromatic prime p > 10^3 such that R(p), in this case 751, is also prime, and the sum of digits sod(p), in this case 1+5+7 = 13, is likewise an emirp (prime with reversal a different prime). a(4) = 1097 becase it is the smallest 4-digit prime, whose reversal (7901) is a different prime, and whose digital sum 1+0+9+7 = 17, which is prime and has a prime reversal (71).
%Y A167992 Cf. A000040, A004086, A007500, A006567, A114018.
%K A167992 base,more,nonn,new
%O A167992 1,3
%A A167992 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2009

%I A167990
%S A167990 0,2,0,3,0,0,4,2,0,0,5,0,0,0,0,6,3,2,0,0,0,7,0,0,0,0,0,0,8,4,0,2,0,0,0,
%T A167990 0,9,0,3,0,0,0,0,0,0,10,5,0,0,2,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,12,6,4,
%U A167990 3,0,2,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,2,0,0,0,0,0
%N A167990 Elements in A126988 (by row) that are not 1.
%C A167990 GCD of rows is A014963.
%e A167990 Table begins:
%e A167990 0,
%e A167990 2,0,
%e A167990 3,0,0,
%e A167990 4,2,0,0,
%e A167990 5,0,0,0,0,
%e A167990 6,3,2,0,0,0,
%e A167990 7,0,0,0,0,0,0,
%e A167990 8,4,0,2,0,0,0,0,
%e A167990 9,0,3,0,0,0,0,0,0,
%e A167990 10,5,0,0,2,0,0,0,0,0,
%e A167990 11,0,0,0,0,0,0,0,0,0,0,
%e A167990 12,6,4,3,0,2,0,0,0,0,0,0,
%Y A167990 Cf. A126988, A014963.
%K A167990 nonn,tabl,new
%O A167990 1,2
%A A167990 Mats Granvik (mats.granvik(AT)abo.fi), Nov 16 2009

%I A167991
%S A167991 2,3,4,4,4,5,6,6,6,6,6,7,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,11,
%T A167991 12,12,12,12,12,12,12,12,12,12,12,13,14,14,14,14,14,14,14,14,14,14,14,
%U A167991 14,14,15,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,17,18,18,18,18
%N A167991 First differences of A167381=1,3,6,10,14,18,23,.
%C A167991 From A020725. Linked to Janet form. Sum by 2n terms is 5,17,37,65,101,145,=4*n^2+1=A053755 for which a recurrence is a(n)=3a(n-1)-3a(n-2)+a(n-3).
%F A167991 2n-1 times 2n followed with 2n+1.
%K A167991 nonn,uned,new
%O A167991 1,1
%A A167991 Paul Curtz (bpcrtz(AT)free.fr), Nov 16 2009

%I A167987
%S A167987 1,63,2766,194650,21086055,3257119761
%N A167987 Number of cycles in the graph of the n-orthoplex, n>=2.
%C A167987 Row sums of Triangle in A167986.
%C A167987 The n-orthoplex, also known as the n-cross-polytope, is the dual of the n-cube.
%H A167987 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/CrossPolytope.html"> Cross Polytope </a>
%e A167987 a(3) = 63, because in dimension n=3, the orthoplex is the octahedron, which has 63 cycles in its graph.
%Y A167987 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%K A167987 more,nonn,new
%O A167987 2,2
%A A167987 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167986
%S A167986 0,1,8,15,24,16,32,102,288,640,960,744,80,370,1584,5920,18240,43080,
%T A167986 69120,56256,160,975,5664,30080,141120,564120,1835520,4542336,7580160,
%U A167986 6385920,280,2121,15624,108080,684480,3876600,19138560,79805376
%N A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.
%C A167986 row n contains 2n-2 elements.
%C A167986 The n-orthoplex is the dual polytope of the n-cube.
%C A167986 The orthoplex is also known as the cross-polytope.
%C A167986 Triangle starts
%C A167986 . 0, 1,
%C A167986 . 8, 15, 24, 16,
%C A167986 . 32, 102, 288, 640, 960, 744,
%C A167986 . 80, 370, 1584, 5920, 18240, 43080, 69120, 56256
%H A167986 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/CrossPolytope.html"> Cross Polytope </a>
%e A167986 T(3,3) = 8, because in dimension n=3, the cross-polytope is the octahedron, which has 8 3-cycles in its graph.
%Y A167986 Cf. A167987 - Row sums of this Triangle.
%Y A167986 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167986 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%K A167986 nonn,tabf,new
%O A167986 2,3
%A A167986 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167985
%S A167985 1200,5400,29520,187200,1310400,9813600,77193600,630538632
%N A167985 Number of n-cycles on the graph of the regular 600-cell, 3<=n<=120.
%C A167985 The 600-cell is one of 6 regular convex polytopes in 4 dimensions. The Schlafli symbol for the 600-cell is {3,3,5}.
%H A167985 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/600-Cell.html"> 600-cell </a>
%H A167985 A. Weimholt <a href="http://www.weimholt.com/andrew/600.html"> 600-cell net </a>
%e A167985 a(3) = 1200, because there are 1200 3-cycles on the graph of the 600-cell.
%Y A167985 Cf. A167981 - 2n-cycles on graph of the tesseract
%Y A167985 Cf. A167982 - n-cycles on graph of 16-cell
%Y A167985 Cf. A167983 - n-cycles on graph of 24-cell
%Y A167985 Cf. A167984 - n-cycles on graph of 120-cell
%Y A167985 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167985 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%Y A167985 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%Y A167985 Cf. A118785 - Number of vertices n-steps from a given vertex on graph of the 600-cell
%K A167985 fini,more,nonn,new
%O A167985 3,1
%A A167985 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167984
%S A167984 0,0,720,0,0,3600,2400,4320,28800,35400,64800,284400,540000,1139400,
%T A167984 3708000,8557200,19677600,55725120,140359200,346456800,935942400
%N A167984 Number of n-cycles on the graph of the regular 120-cell, 3<=n<=600.
%C A167984 The 120-cell is one of 6 regular convex polytopes in 4 dimensions. The Schlafli symbol of the 120-cell is {5,3,3}.
%H A167984 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/120-Cell.html"> 120-cell </a>
%H A167984 A. Weimholt <a href="http://www.weimholt.com/andrew/120.html"> 120-cell net </a>
%e A167984 a(5) = 720, because there are 720 5-cycles on the graph of the 120-cell.
%Y A167984 Cf. A167981 - 2n-cycles on graph of the tesseract
%Y A167984 Cf. A167982 - n-cycles on graph of 16-cell
%Y A167984 Cf. A167983 - n-cycles on graph of 24-cell
%Y A167984 Cf. A167985 - n-cycles on graph of 600-cell
%Y A167984 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167984 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%Y A167984 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%Y A167984 Cf. A108997 - Number of vertices n-steps from a given vertex on graph of 120-cell
%K A167984 fini,more,nonn,new
%O A167984 3,3
%A A167984 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167983
%S A167983 96,360,1440,7120,37728,196488,974592,4536000,19934208,82689264,322437312,
%T A167983 1171745280,3924079104,11964375936,32761139328,79244294016,165800420352,
%U A167983 291640320576,413774810112,443415854592,318534709248,114869295744
%N A167983 Number of n-cycles on the graph of the regular 24-cell, 3<=n<=24.
%C A167983 The 24-cell is one of 6 regular convex polytopes in 4 dimensions. The Schlafli symbol of the 24-cell is {3,4,3}.
%H A167983 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/24-Cell.html"> 24-cell </a>
%H A167983 A. Weimholt <a href="http://www.weimholt.com/andrew/24.html"> 24-cell net </a>
%H A167983 Max Alekseyev, <a href="http://cse.sc.edu/~maxal/gpscripts/">PARI/GP scripts for various math problems</a>
%e A167983 a(3) = 96, because there are 96 3-cycles on the graph of the 24-cell.
%Y A167983 Cf. A167981 - 2n-cycles on graph of the tesseract
%Y A167983 Cf. A167982 - n-cycles on graph of 16-cell
%Y A167983 Cf. A167984 - n-cycles on graph of 120-cell
%Y A167983 Cf. A167985 - n-cycles on graph of 600-cell
%Y A167983 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167983 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%Y A167983 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%K A167983 fini,full,nonn,new
%O A167983 3,1
%A A167983 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009
%E A167983 a(16)-a(24) and "full" keyword from Max Alekseyev (maxale(AT)gmail.com), Nov 18 2009

%I A167982
%S A167982 32,102,288,640,960,744
%N A167982 Number of n-cycles on the graph of the regular 16-cell, 3<=n<=8.
%C A167982 Row n=3 of the triangle in A167986
%C A167982 The 16-cell is the dual polytope of the tesseract, and is one of 6 regular convex polytopes in 4 dimensions. The Schlafli symbol for the 16-cell is {3,3,4}.
%H A167982 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/16-Cell.html"> 16-cell </a>
%H A167982 A. Weimholt <a href="http://www.weimholt.com/andrew/16.html"> 16-cell net </a>
%e A167982 a(3) = 32, because there are 32 3-cycles on the graph of the 16-cell.
%Y A167982 Cf. A167981 - 2n-cycles on graph of the tesseract
%Y A167982 Cf. A167983 - n-cycles on graph of 24-cell
%Y A167982 Cf. A167984 - n-cycles on graph of 120-cell
%Y A167982 Cf. A167985 - n-cycles on graph of 600-cell
%Y A167982 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167982 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%Y A167982 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%K A167982 fini,full,nonn,new
%O A167982 3,1
%A A167982 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167981
%S A167981 24,128,696,2112,5024,5736,1344
%N A167981 Number of 2n-cycles on the graph of the tesseract, 2<=n<=8
%C A167981 Row n=4 of the triangle in A085452
%C A167981 The graph of any n-cube (n>1) contains only even length cycles.
%C A167981 The tesseract is the 4 dimensional cube, and is one of 6 regular convex polytopes in 4 dimensions. The Schlafli symbol for the tesseract is {4,3,3}.
%H A167981 Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/Tesseract.html">tesseract</a>
%H A167981 A. Weimholt <a href="http://www.weimholt.com/andrew/tesseract.html">tesseract net</a>
%e A167981 a(2) = 24 because there are 24 4-cycles on the graph of the tesseract.
%Y A167981 Cf. A167982 - n-cycles on graph of 16-cell
%Y A167981 Cf. A167983 - n-cycles on graph of 24-cell
%Y A167981 Cf. A167984 - n-cycles on graph of 120-cell
%Y A167981 Cf. A167985 - n-cycles on graph of 600-cell
%Y A167981 Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube
%Y A167981 Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.
%Y A167981 Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex
%K A167981 fini,full,nonn,new
%O A167981 2,1
%A A167981 Andrew Weimholt (andrew(AT)weimholt.com), Nov 16 2009

%I A167979
%S A167979 1,2,6,3,10,12,4,13,20,14,5,18,25,22,24,7,21,36,29,40,26,8,27,41,38,49,
%T A167979 42,28,9,34,51,45,72,53,44,30,11,37,68,59,81,74,57,46,48,15,43,73,70,99,
%U A167979 85,76,61,80,50,16,55,83,77,136,107,89,8,97,82,52
%N A167979 Linearize the arrays A099627 A124922 ... defined in A167204 and based on Tabf A161924 then concatenate to form a new Table.
%e A167979 The resulting table begins:
%e A167979 ..1..2..3..4..5..7..8
%e A167979 ..6.10.13.18.21.27
%e A167979 .12.20.25.36.41
%e A167979 .14.22.29.38
%e A167979 etc.
%Y A167979 Cf. A099627 A124922 A167204 A161924
%K A167979 nonn,tabl,uned,new
%O A167979 1,2
%A A167979 Alford Arnold (Alfprd1940(AT)aol.com), Nov 15 2009

%I A167976
%S A167976 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A167976 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,1,48,2,
%U A167976 49,3,50,4,51,5,52,6,53,7,54,8,55,9,56,10,57,11,58,12,59,13,60,14,61,15
%N A167976 Signature sequence of Phi^8 = 46.978713763748..., where Phi is the golden ratio 1.6180339887499... .
%D A167976 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167976 Cf. A084531, A084532
%K A167976 nonn,new
%O A167976 1,2
%A A167976 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167974
%S A167974 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,1,19,2,20,3,21,4,22,5,23,
%T A167974 6,24,7,25,8,26,9,27,10,28,11,29,12,30,13,31,14,32,15,33,16,34,17,35,18,
%U A167974 36,1,19,37,2,20,38,3,21,39,4,22,40,5,23,41,6,24,42,7,25,43,8,26,44,9
%N A167974 Signature sequence of Phi^6 = 17.944271909999..., where Phi is the golden ratio 1.6180339887499... .
%D A167974 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167974 Cf. A084531, A084532
%K A167974 nonn,new
%O A167974 1,2
%A A167974 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167968
%S A167968 1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,
%T A167968 1,3,2,1,3,2,1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,1,
%U A167968 5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1,5,4,3,2,1
%N A167968 Signature sequence of phi^4 = 0.14589803375032..., where phi is the golden ratio 0.61803398874989... .
%D A167968 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167968 Cf. A084531, A084532
%K A167968 nonn,new
%O A167968 1,8
%A A167968 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167964
%S A167964 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A167964 1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,
%U A167964 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1
%N A167964 Signature sequence of phi^8 = 0.021286236252208..., where phi is the golden ratio 0.61803398874989... .
%D A167964 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167964 Cf. A084531, A084532
%K A167964 nonn,new
%O A167964 1,48
%A A167964 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167972
%S A167972 1,2,3,4,5,6,7,1,8,2,9,3,10,4,11,5,12,6,13,7,14,1,8,15,2,9,16,3,10,17,4,
%T A167972 11,18,5,12,19,6,13,20,7,14,21,1,8,15,22,2,9,16,23,3,10,17,24,4,11,18,
%U A167972 25,5,12,19,26,6,13,20,27,7,14,21,28,1,8,15,22,29,2,9,16,23,30,3,10,17
%N A167972 Signature sequence of Phi^4 = 6.8541019662497..., where Phi is the golden ratio 1.6180339887499... .
%D A167972 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167972 Cf. A084531, A084532
%K A167972 nonn,new
%O A167972 1,2
%A A167972 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167975
%S A167975 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A167975 27,28,29,30,1,31,2,32,3,33,4,34,5,35,6,36,7,37,8,38,9,39,10,40,11,41,
%U A167975 12,42,13,43,14,44,15,45,16,46,17,47,18,48,19,49,20,50,21,51,22,52,23
%N A167975 Signature sequence of Phi^7 = 29.034441853749..., where Phi is the golden ratio 1.6180339887499... .
%D A167975 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167975 Cf. A084531, A084532
%K A167975 nonn,new
%O A167975 1,2
%A A167975 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167973
%S A167973 1,2,3,4,5,6,7,8,9,10,11,12,1,13,2,14,3,15,4,16,5,17,6,18,7,19,8,20,9,
%T A167973 21,10,22,11,23,12,1,24,13,2,25,14,3,26,15,4,27,16,5,28,17,6,29,18,7,30,
%U A167973 19,8,31,20,9,32,21,10,33,22,11,34,23,12,1,35,24,13,2,36,25,14,3,37,26
%N A167973 Signature sequence of Phi^5 = 11.090169943749..., where Phi is the golden ratio 1.6180339887499... .
%D A167973 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167973 Cf. A084531, A084532
%K A167973 nonn,new
%O A167973 1,2
%A A167973 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167969
%S A167969 1,1,1,1,1,2,1,2,1,2,1,2,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,4,1,2,3,4,1,2,3,
%T A167969 4,1,2,3,4,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,6,2,3,4,
%U A167969 5,1,6,2,3,4,5,1,6,2,3,4,5,1,6,2,3,4,5,1,6,2,7,3,4,5,1,6,2,7,3,4,5,1,6
%N A167969 Signature sequence of phi^3 = 0.23606797749979..., where phi is the golden ratio 0.61803398874989... .
%D A167969 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167969 Cf. A084531, A084532
%K A167969 nonn,new
%O A167969 1,6
%A A167969 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167971
%S A167971 1,2,3,4,5,1,6,2,7,3,8,4,9,5,1,10,6,2,11,7,3,12,8,4,13,9,5,1,14,10,6,2,
%T A167971 15,11,7,3,16,12,8,4,17,13,9,5,1,18,14,10,6,2,19,15,11,7,3,20,16,12,8,4,
%U A167971 21,17,13,9,5,22,1,18,14,10,6,23,2,19,15,11,7,24,3,20,16,12,8,25,4,21
%N A167971 Signature sequence of Phi^3 = 4.2360679774998..., where Phi is the golden ratio 1.6180339887499... .
%D A167971 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167971 Cf. A084531, A084532
%K A167971 nonn,new
%O A167971 1,2
%A A167971 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167970
%S A167970 1,1,1,2,1,2,1,2,1,3,2,1,3,2,1,3,2,4,1,3,2,4,1,3,2,4,1,3,5,2,4,1,3,5,2,
%T A167970 4,1,3,5,2,4,1,6,3,5,2,4,1,6,3,5,2,4,1,6,3,5,2,7,4,1,6,3,5,2,7,4,1,6,3,
%U A167970 5,2,7,4,1,6,3,8,5,2,7,4,1,6,3,8,5,2,7,4,1,6,3,8,5,2,7,4,9,1,6,3,8,5,2
%N A167970 Signature sequence of phi^2 = 0.38196601125011..., where phi is the golden ratio 0.61803398874989... .
%D A167970 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167970 Cf. A084531, A084532
%K A167970 nonn,new
%O A167970 1,4
%A A167970 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167967
%S A167967 1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,
%T A167967 3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,4,
%U A167967 1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,1,2,3
%N A167967 Signature sequence of phi^5 = 0.090169943749474..., where phi is the golden ratio 0.61803398874989... .
%D A167967 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167967 Cf. A084531, A084532
%K A167967 nonn,new
%O A167967 1,13
%A A167967 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167966
%S A167966 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,
%T A167966 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,
%U A167966 2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1
%N A167966 Signature sequence of phi^6 = 0.055728090000841..., where phi is the golden ratio 0.61803398874989... .
%D A167966 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167966 Cf. A084531, A084532
%K A167966 nonn,new
%O A167966 1,19
%A A167966 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167965
%S A167965 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,
%T A167965 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,
%U A167965 2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3
%N A167965 Signature sequence of phi^7 = 0.034441853748633..., where phi is the golden ratio 0.61803398874989... .
%D A167965 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%Y A167965 Cf. A084531, A084532
%K A167965 nonn,new
%O A167965 1,31
%A A167965 Casey Mongoven (cm(AT)caseymongoven.com), Nov 15 2009

%I A167963
%S A167963 13,113,1009,10007,100049
%N A167963 Least emirp (A006567) greater than 10^n.
%C A167963 Least emirp (nonpalindromatic prime in A007500, i.e. prime whose reversal is a different prime) greater than 10^n.
%F A167963 a(n) = Min{p > 10^n in A006567} = Min{p > 10^n in A000040 such that A004086(p) is in A000040, and A004086(p) distinct from p}.
%e A167963 a(1) = 13 because 13 is the least nonpalindromatic prime p > 10^1 such that R(p), in this case 31, is also prime (if we allow palindromes, then 11 would work). a(2) = 113 because 113 is the least nonpalindromatic prime p > 10^2 such that R(p), in this case 311, is also prime (if we allow palindromes, then 101 would work).
%Y A167963 Cf. A000040, A004086, A007500, A006567, A114018.
%K A167963 base,more,nonn,new
%O A167963 1,1
%A A167963 Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 15 2009

%I A167948
%S A167948 1,0,1,0,1,1,0,0,1,2,0,0,1,2,3,0,0,0,2,3,6,0,0,0,2,3,6,11,0,0,0,0,3,6,
%T A167948 11,22,0,0,0,0,3,6,11,22,42,0,0,0,0,0,6,11,22,42,84,0,0,0,0,0,6,11,22,
%U A167948 42,84,165,0,0,0,0,0,0,11,22,42,84,165,330
%N A167948 Triangle read by rows, A101688 * (an infinite lower triangular matrix with A002083 as the main diagonal and the rest zeros).
%C A167948 Row sums = A002083(n+1): (1, 1, 2, 3, 6, 11, 22, 42, 84, 165,...). Sum of n-th row terms = rightmost term of next row.
%C A167948 Eigensequence of triangle A101688 = A002083 starting with offset 1:
%C A167948 (1, 1, 2, 3, 6, 11, 22, 42,...).
%F A167948 Triangle A167948 = M * Q, M = triangle A101688, Q = an infinite lower triangular
%F A167948 matrix with (1, 1, 1, 2, 3, 6, 11, 22, 42,...) as the main diagonal and the
%F A167948 rest zeros.
%e A167948 First few rows of the triangle = \KQ 1;
%e A167948 0, 1;
%e A167948 0, 1, 1;
%e A167948 0, 0, 1, 2;
%e A167948 0, 0, 1, 2, 3;
%e A167948 0, 0, 0, 2, 3, 6;
%e A167948 0, 0, 0, 2, 3, 6, 11;
%e A167948 0, 0, 0, 0, 3, 6, 11, 22;
%e A167948 0, 0, 0, 0, 3, 6, 11, 22, 42;
%e A167948 0, 0, 0, 0, 0, 6, 11, 22, 42, 84;
%e A167948 0, 0, 0, 0, 0, 6, 11, 22, 42, 84, 165;
%e A167948 0, 0, 0, 0, 0, 0, 11, 22, 42, 84, 165, 330;
%e A167948 0, 0, 0, 0, 0, 0, 11, 22, 42, 84, 165, 330, 654;
%e A167948 0, 0, 0, 0, 0, 0, .0, 22, 42, 84, 165, 330, 654, 1308;
%e A167948 0, 0, 0, 0, 0, 0, .0, 22, 42, 84, 165, 330, 654, 1308, 2605;
%e A167948 ...
%Y A167948 Cf. A101688, A002083
%K A167948 nonn,tabl,new
%O A167948 1,10
%A A167948 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2009

%I A160743
%S A160743 0,8,17593,389112,3169562,15694600,57385803,170880248,438565492,1005601032,
%T A160743 2110507325,4124403448,7599974478,13331249672,22425272527,36386743800,57216718568,
%U A160743 87526438408,130667379777,190878599672,273452459650,384919809288,533255710163
%N A160743 8*LegendreP[7,n] (using Mma's notation).
%t A160743 Table[8 LegendreP[7,n],{n,0,50}]
%K A160743 nonn,new
%O A160743 0,2
%A A160743 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A160741
%S A160741 5,10159,867211,10373071,59271739,227860495,683245579,1727242351,3854919931,
%T A160741 7823790319,14733641995,26117017999,44040338491,71215667791,111123125899,
%U A160741 168143944495,247704167419,356428995631,502307776651,694869638479,945369767995
%V A160741 -5,10159,867211,10373071,59271739,227860495,683245579,1727242351,3854919931,
%W A160741 7823790319,14733641995,26117017999,44040338491,71215667791,111123125899,
%X A160741 168143944495,247704167419,356428995631,502307776651,694869638479,945369767995
%N A160741 Numerator of LegendreP[6,2n] (using Mma's notation).
%t A160741 Table[Numerator[LegendreP[6,2n]],{n,0,40}]
%Y A160741 Cf. A160739, A144126.
%K A160741 sign,new
%O A160741 0,1
%A A160741 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A160739
%S A160739 5,16,10159,143824,867211,3415120,10373071,26425744,59271739,120704656,
%T A160739 227860495,404631376,683245579,1106013904,1727242351,2615311120,3854919931,
%U A160739 5549499664,7823790319,10826585296,14733641995,19750758736,26117017999
%V A160739 -5,16,10159,143824,867211,3415120,10373071,26425744,59271739,120704656,
%W A160739 227860495,404631376,683245579,1106013904,1727242351,2615311120,3854919931,
%X A160739 5549499664,7823790319,10826585296,14733641995,19750758736,26117017999
%N A160739 16*LegendreP[6,n] (using Mma's notation).
%t A160739 Table[16 LegendreP[6,n],{n,0,40}]
%Y A160739 Cf. A144126, A160741.
%K A160739 sign,new
%O A160739 0,1
%A A160739 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A160737
%S A160737 0,4,743,6732,30046,94100,237429,517468,1014332,1834596,3115075,5026604,
%T A160737 7777818,11618932,16845521,23802300,32886904,44553668,59317407,77757196,
%U A160737 100520150,128325204,161966893,202319132,250338996,307070500,373648379
%N A160737 4*LegendreP[5,n] (using Mma's notation).
%t A160737 Table[4 LegendreP[5,n],{n,0,50}]
%K A160737 nonn,new
%O A160737 0,2
%A A160737 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A160699
%S A160699 0,17,154,531,1268,2485,4302,6839,10216,14553,19970,26587,34524,43901,54838,
%T A160699 67455,81872,98209,116586,137123,159940,185157,212894,243271,276408,312425,
%U A160699 351442,393579,438956,487693,539910,595727,655264,718641,785978,857395
%N A160699 A bisection of A063522.
%t A160699 Table[LegendreP[3,2n],{n,0,50}]
%K A160699 nonn,new
%O A160699 0,2
%A A160699 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A160674
%S A160674 1,63,305,847,1809,3311,5473,8415,12257,17119,23121,30383,39025,49167,60929,
%T A160674 74431,89793,107135,126577,148239,172241,198703,227745,259487,294049,331551,
%U A160674 372113,415855,462897,513359,567361,625023,686465,751807,821169,894671
%N A160674 A bisection of A063522.
%t A160674 Table[LegendreP[3,2n+1],{n,0,50}]
%K A160674 nonn,new
%O A160674 0,2
%A A160674 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A140870
%S A140870 3,443,8483,44283,141443,347003,721443,1338683,2286083,3664443,5588003,
%T A140870 8184443,11594883,15973883,21489443,28323003,36669443,46737083,58747683,
%U A140870 72936443,89552003,108856443,131125283,156647483,185725443,218675003,255825443
%N A140870 8*LegendreP[4,2n] (using Mma's notation).
%C A140870 I would like to know how to compute this in Maple!
%t A140870 Table[8 LegendreP[4,2n],{n,0,50}]
%o A140870 (MAGMA code from Klaus Brockhaus, Nov 18 2009)
%o A140870 P<x> := PolynomialRing(IntegerRing());
%o A140870 LP4:=LegendrePolynomial(4);
%o A140870 [ Evaluate(8*LP4, 2*n): n in [0..26] ];
%Y A140870 Cf. A144124.
%K A140870 nonn,new
%O A140870 0,1
%A A140870 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2009

%I A167939
%S A167939 1,3,10,64,973,31743,2069970,267270040,68629753649,35171000942707,
%T A167939 36024807353574290,73784587576805254664,302228602363365451957805,
%U A167939 2475873310144021668263093215,40564787336902311168400640561098
%N A167939 The number of connected subgraphs of the complete graph with n nodes.
%C A167939 The problem originated from Attila Szabss^3 .
%e A167939 For n = 3, consider the complete graph with nodes A, B and C. a(3) = 10, the 10 complete subgraphs are: A, B, C, AB, AC, BC, AB+AC, AB+BC, AC+BC, AB+AC+BC.
%o A167939 (Other) -- Haskell
%o A167939 import Data.Function (fix)
%o A167939 import Data.List (transpose)
%o A167939 a :: [Integer]
%o A167939 a = scanl1 (+) . (!! 1) . transpose . fix $ map ((1:) . zipWith (*) (scanl1 (*) l) . zipWith poly (scanl1 (+) l)) . scanl (flip (:)) [] . zipWith (zipWith (*)) pascal where l = iterate (2*) 1
%o A167939 -- the Pascal triangle
%o A167939 pascal :: [[Integer]]
%o A167939 pascal = iterate (\l -> zipWith (+) (0: l) l) (1: repeat 0)
%o A167939 -- evaluate a polynom at a given value
%o A167939 poly :: (Num a) => a -> [a] -> a
%o A167939 poly t = foldr (\e i -> e + t*i) 0
%K A167939 full,nonn,new
%O A167939 1,2
%A A167939 Peter Divianszky (divip(AT)aszt.inf.elte.hu), Nov 15 2009

%I A167932
%S A167932 1,1,2,3,4,4,7,6,9,10,13,13,20
%N A167932 Number of partitions of n such that all parts are equal or all parts are distinct.
%C A167932 Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).
%H A167932 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>
%H A167932 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A167932 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A167932 a(n) = A000041(n) - A167930(n).
%F A167932 a(n) = A000009(n) + A032741(n).
%e A167932 The partitions of 6 are:
%e A167932 6 ....................... All parts are distinct ......... (1).
%e A167932 5 + 1 ................... All parts are distinct ......... (2).
%e A167932 4 + 2 ................... All parts are distinct ......... (3).
%e A167932 4 + 1 + 1 ............... Only some parts are equal.
%e A167932 3 + 3 ................... All parts are equal ............ (4).
%e A167932 3 + 2 + 1 ............... All parts are distinct ......... (5).
%e A167932 3 + 1 + 1 + 1 ........... Only some parts are equal.
%e A167932 2 + 2 + 2 ............... All parts are equal ............ (6).
%e A167932 2 + 2 + 1 + 1 ........... Only some parts are equal.
%e A167932 2 + 1 + 1 + 1 + 1 ....... Only some parts are equal.
%e A167932 1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal ............ (7).
%e A167932 Then a(6) = 7.
%Y A167932 Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.
%K A167932 more,nonn,new
%O A167932 0,3
%A A167932 Omar E. Pol (info(AT)polprimos.com), Nov 15 2009

%I A167936
%S A167936 0,1,1,5,7,23,37,101,175,431,781,1805,3367,7463,14197
%N A167936 a(n)=2a(n-1)+3a(n-2)-6a(n-3). Valuable for successive differences:first is A167784. Also first differences of A167762. Also binomial transform of (O,A077917 or 0,A127864 signed)=0,1,-1,5,-11,33,-87,.
%C A167936 From a(n)=p*a(n-1)+q*a(n-2)-p*q*a(n-3).See submitted A167910. From Poly-Bernoulli numbers,via A085350.
%F A167936 Mix A005061 , A085350. a(n)=2a(n-1)+(1,-1,3,-3,9,-9,=A108411).
%K A167936 nonn,uned,new
%O A167936 0,4
%A A167936 Paul Curtz (bpcrtz(AT)free.fr), Nov 15 2009

%I A167921
%S A167921 1,3,5,11,17,22,29,36,41,46,52,59,66,71,78,82,88,101,107,112,126,130,
%T A167921 137,149,156,162,166,172,179,191,197,210,222,227,232,239,250,256,262,
%U A167921 269,276,281,292,306,311,316,330,336,347,352,358,366,372,378,382,388
%N A167921 Single or isolated numbers-1.
%Y A167921 Cf. A167706.
%K A167921 nonn,new
%O A167921 1,2
%A A167921 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 15 2009

%I A167930
%S A167930 0,0,0,0,1,3,4,9,13,20,29,43,57
%N A167930 Number of partitions of n such that only some parts are equal.
%H A167930 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>
%H A167930 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A167930 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A167930 a(n) = A047967(n) - A032741(n).
%F A167930 a(n) = A000041(n) - A000009(n) - A032741(n).
%F A167930 a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.
%e A167930 The partitions of 6 are:
%e A167930 6 ....................... All parts are distinct.
%e A167930 5 + 1 ................... All parts are distinct.
%e A167930 4 + 2 ................... All parts are distinct.
%e A167930 4 + 1 + 1 ............... Only some parts are equal ...... (1).
%e A167930 3 + 3 ................... All parts are equal.
%e A167930 3 + 2 + 1 ............... All parts are distinct.
%e A167930 3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
%e A167930 2 + 2 + 2 ............... All parts are equal.
%e A167930 2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
%e A167930 2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
%e A167930 1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
%e A167930 Then a(6) = 4.
%Y A167930 Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.
%K A167930 more,nonn,new
%O A167930 0,6
%A A167930 Omar E. Pol (info(AT)polprimos.com), Nov 15 2009
%E A167930 Edited by Omar E. Pol (info(AT)polprimos.com), Nov 16 2009

%I A167920
%S A167920 2,37,37,37,6,37,113,89,37,131,12,37,53,113,211,113,18,37,457,401,127,
%T A167920 23,47,337,251,53,163,113,30,211,373,257,67,307,211,37,223,457,79,401,
%U A167920 42,127,173,89,541,47,941,337,197,251,307,53,743,541,331,113,457,233,60
%N A167920 Smallest single or isolated number==1(mod n).
%e A167920 If n=72, the smallest single or isolated number in the sequence 73, 145, 217, 289, 361, 433, 505, 577, so a(72)=577.
%Y A167920 Cf. A034694, A167706.
%K A167920 nonn,new
%O A167920 1,1
%A A167920 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 15 2009

%I A167925
%S A167925 0,1,1,1,2,3,0,2,6,12,1,0,9,32,75,1,4,9,80,275,684,0,8,0,192,1000,
%T A167925 3240,8232,1,8,27,448,3625,15336,47677,122368,1,0,81,1024,13125,
%U A167925 72576,276115,835584,2158569,0,16,162,2304,47500,343440,1599066
%V A167925 0,1,1,1,2,3,0,2,6,12,-1,0,9,32,75,-1,-4,9,80,275,684,0,-8,0,192,1000,
%W A167925 3240,8232,1,-8,-27,448,3625,15336,47677,122368,1,0,-81,1024,13125,
%X A167925 72576,276115,835584,2158569,0,16,-162,2304,47500,343440,1599066
%N A167925 A triangular sequence of the Matrix Markov type based on the 2x2 matrix: m={{a,1},{-1,1}}; which has determinant equal to trace.
%C A167925 Row sums are:
%C A167925 {0, 2, 6, 20, 115, 1043, 12656, 189420, 3356913, 68661516, 1591360540,...}
%C A167925 Each row is a specific Markov sequence with a different limiting ratio.
%e A167925 {0},
%e A167925 {1, 1},
%e A167925 {1, 2, 3},
%e A167925 {0, 2, 6, 12},
%e A167925 {-1, 0, 9, 32, 75},
%e A167925 {-1, -4, 9, 80, 275, 684},
%e A167925 {0, -8, 0, 192, 1000, 3240, 8232},
%e A167925 {1, -8, -27, 448, 3625, 15336, 47677, 122368},
%e A167925 {1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569},
%e A167925 {0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000},
%e A167925 {-1, 32, -243, 5120, 171875, 1625184, 9260657, 38961152, 133155495, 390500000, 1017681269}
%t A167925 Clear[m, a, n, v];
%t A167925 m = {{a, 1}, {-1, 1}};
%t A167925 v[0] := {0, 1};
%t A167925 v[n_] := v[n] = m.v[n - 1];
%t A167925 Table[v[n][[1]], {n, 0, 10}, {a, 0, n}];
%t A167925 Flatten[%]
%K A167925 sign,uned,new
%O A167925 0,5
%A A167925 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 15 2009

%I A167918
%S A167918 6,16,5,4,17,10,20,13,55,17,26,44,81,41,35,102,30,162,33,61,49,66,173,
%T A167918 42,45,127,65,66,228,52
%N A167918 a(n) is smallest index k > n of k-th prime with f(n,k):=(p(k)+p(k+1))/(p(n)+p(n+1)) an integer >=2 (n=1,2,...)
%C A167918 (1) It is conjectured that sequence is infinite
%C A167918 (2) It is conjectured that f(n,k)=2 for infinite many cases
%C A167918 (3) Note the new link between two consecutive primes and prime twins
%C A167918 (4) Note many possible generalizations with other fraction types (p(k)+...+p(k+s))/(p(n)+...+p(n+t))
%C A167918 (5) Open problems: (a) is f(n,k) bounded, (b) which integer values for f(n,k) are "possible"
%D A167918 Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer 2005
%D A167918 Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
%D A167918 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005)
%e A167918 (1) f(1,6)=(p(6)+p(7))/(p(1)+p(2))=(13+17)/(2+3)=6 gives a(1)=6
%e A167918 (2) f(18,162)=(p(162)+p(163))/(p(18)+p(19))=(953+967)/(61+67)=15 gives a(18)=162
%Y A167918 Cf. A000040 The prime numbers
%Y A167918 Cf. A167790
%K A167918 nonn,new
%O A167918 1,1
%A A167918 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 15 2009

%I A167917
%S A167917 3,7,31,8191,524287
%N A167917 Mersenne primes that belong to Cunningham chains = {3, 7} U {Mp | 2Mp - 1 is prime. (Mp a Mersenne prime)}.
%C A167917 If p is prime, p >= 5, and Mp belongs to a chain, Mp is always the first term of a chain of the second kind. This is true since (Mp+1)/2 = (2^p - 1 +1)/2 = 2^(p-1), which is composite for p >= 3. (Mp-1)/2 = (2^p - 1 -1)/2 = 2^(p-1)-1 = a. For p >= 5, a is composite since a>3, and a mod 3 = 0. Finally 2Mp + 1 = 2(2^p - 1)+1 = 2^(p+1)-1 = a. If p>=3, a is composite because a > 3, and a mod 3 = 0. We can conclude that beginning with 31, a Mersenne prime can only starts a Cunningham chain of the second kind. If Mp >= 31 starts a chain, the second term of this chain is 2Mp -1=2(2^p - 1)-1 = 2^(p+1) - 3.
%C A167917 That is a number of the form 2^N - 3, even N, so also of the form a^2 - 3, a = 2^(N/2). In this case any factor f of the second term of a chain satisfies f mod 24=1, or f mod 24=11, or f mod 24=13, or f mod 24=23. (1) The next term of this sequence is an unknown Mersenne prime. Probably many primes of this kind will be determined until this term be found. In the work with the known Mersenne primes, M42643801 gives T=2^(42643801+1) -3. The smallest factor of T is f = 38334482051, which is greater than 2^35.
%C A167917 Considering the probabilities given in the second reference, one can conclude that before T was identified as composed (by the exam of all the primes less than f satisfying (1)), the probability of prime T reached a value of 1 in 609,197. This probability is small, but not negligible. Note that the largest known Cunningham chain of length 2 has starting prime 607095* 2^176311 - 1. This is a "very small chain" compared with a chain beginning with a new Mersenne prime.
%H A167917 Mathpages, <a href="http://www.mathpages.com/home/kmath480.htm">Some Properties of the Lucas Sequence</a>
%H A167917 GIMPS, <a href="http://www.mersenne.org/various/math.php">Mathematics and Research Strategy</a>
%H A167917 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>
%e A167917 a(1) = 3 since 2*3 - 1 = 5. a(2) = 7 because 2*7 - 1 = 13.
%Y A167917 Cf. A000043, A000668, A050415, A050414.
%K A167917 hard,nonn,new
%O A167917 1,1
%A A167917 W. Bomfim (webonfim(AT)bol.com.br), Nov 15 2009

%I A167915
%S A167915 5,17,19,29,31,41,43,53,67,71,79,89,97,101,103,109,113,127,131,137,139,
%T A167915 149,151,163,173,181,191,197,199,211,223,229,233,239,241,251,257,269,
%U A167915 271,281,283,293,307,311,317,331,337,349,353,367,373,379,389,401,409
%N A167915 Primes which are the sume of two consecutive nonprimes A141468.
%C A167915 Five together with primes are the sum of two consecutive composite numbers.
%e A167915 a(1)=1+4=5, a(2)=8+9=17.
%Y A167915 Cf. A000040, A060254, A141468.
%K A167915 nonn,new
%O A167915 1,1
%A A167915 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 15 2009

%I A167911
%S A167911 1,2,2,2,1,1,1,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,2,1,
%T A167911 1,1,2,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,2,1,1,
%U A167911 1,1,2,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1
%N A167911 Differences between consecutive non-single (or nonisolated) numbers A167707.
%e A167911 a(10=1-0=1, a(2)=3-1=2, a(3)=5-3=2, a(4)=7-5=2, a(5)=8-7=1.
%Y A167911 Cf. A167707.
%K A167911 nonn,new
%O A167911 1,2
%A A167911 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 15 2009

%I A167913
%S A167913 0,1,2,9,64,25,216,49,4096,6561,10,161051,1728,13,2744,759375,65536,289,
%T A167913 104976,6859,400,4084101,10648,6436343
%N A167913 a(n) = Product of similar consecutive values of A166724(n).
%Y A167913 Cf. A166724.
%K A167913 nonn,new
%O A167913 1,3
%A A167913 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Nov 15 2009

%I A167912
%S A167912 1,217,913083596083
%N A167912 1/(3^n)^2 * Sum[ Binomial[2k,k], {k, 0, 3^n-1} ].
%C A167912 Note that Mod[a(n),27] = Mod[a(n),9] = Mod[a(n),3] = 1.
%H A167912 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>.
%H A167912 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>.
%Y A167912 Cf. A006134, A083096, A066796, A083097, A081601, A010060, A122485.
%K A167912 more,nonn,new
%O A167912 1,2
%A A167912 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 15 2009

%I A167909
%S A167909 2,2,6,6,5,7,7,5,5,6,7,7,5,7,4,6,13,6,5,14,4,7,12,7,6,4,6,7,12,6,13,12,
%T A167909 5,5,7,11,6,6,7,7,5,11,14,5,5,14,6,11,5,6,8,6,6,4,6,8,4,8,11,12,7,4,6,8,
%U A167909 5,5,12,8,4,8,4,6,13,19,6,19,6,7,7,19,6,7,7,6,5,13,11,5,6,7,13,4,6,8,10
%N A167909 Differences between consecutive single (or isolated) numbers A167706.
%e A167909 a(1)=4-2=2, a(2)=6-4=2, a(3)=12-6=6.
%Y A167909 Cf. A167706.
%K A167909 nonn,new
%O A167909 1,1
%A A167909 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 15 2009

%I A167910
%S A167910 0,0,1,3,13,39,133,399,1261,3783,11605,34815
%N A167910 a(n)=3a(n-1)+4a(n-2)-12a(n-3); a(0)=a(1)=0,a(2)=1.
%C A167910 Principal sequence.See A138587. From general recurrence a(n)=p*a(n-1)+q*a(n-2)-p*q*a(n-3). Here p=3,q=4. See A167762 and A167784: p=2,q=3; A167889: p=4,q=9.Also A005061 and A085350 (from Poly-Bernoulli numbers):p=4,q=9; A027649 Poly-Bernoulli numbers: p=3,q=4. a(2n+1)=3*a(2n).
%F A167910 a(n+1)-3a(n)=0,1,0,4,0,16,0,64,= mix A000004,A000302.
%K A167910 nonn,uned,new
%O A167910 0,4
%A A167910 Paul Curtz (bpcrtz(AT)free.fr), Nov 15 2009

%I A140969
%S A140969 11,13,173,191,223,239,251,2731,2749,2767,2797,3019,3023,3037,3067,3259,
%T A140969 3307,3323,3499,3517,3533,3547,3581,3583,3803,3821,3823,4013,4027,4079,
%U A140969 4091,4093,43691,43711,43759,43951,43963,43997,44027,44029,44203,44207
%N A140969 Prime numbers whose hexadecimal representation uses only the digits A,B,C,D,E,F (and not the decimal digits).
%H A140969 Gil Broussard (gilbroussard(AT)bellsouth.net), Jul 27 2008, <a href="b140969.txt">Table of n, a(n) for n = 1..682</a>
%e A140969 11=B 13=D 113=AD 131=BB
%K A140969 base,nonn,word,new
%O A140969 1,1
%A A140969 Gil Broussard (gilbroussard(AT)bellsouth.net), Jul 27 2008
%E A140969 Edited by njas, Nov 15 2009

%I A050216
%S A050216 2,2,5,6,15,9,22,11,27,47,16,57,44,20,46,80,78,32,90,66,30,106,75,114,
%T A050216 163,89,42,87,42,100,354,99,165,49,299,58,182,186,128,198,195,76,356,
%U A050216 77,144,75,463,479,168,82,166,270,90,438,275,274,292,91,292,199,99
%N A050216 Number of primes between (prime(n))^2 and (prime(n+1))^2, with a(0) = 2 by convention.
%C A050216 The function in Brocard's Conjecture.
%C A050216 The lines in the graph correspond to prime gaps of 2, 4, 6,... - T. D. Noe, Feb 04 2008
%H A050216 T. D. Noe, <a href="b050216.txt">Table of n, a(n) for n=0..10000</a>
%H A050216 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BrocardsConjecture.html">Brocard's Conjecture</a>
%e A050216 There are 2 primes less than 2^2, there are 2 primes between 2^2 and 3^2, 5 primes between 5^2 and 7^2, etc.
%t A050216 PrimePi[ Prime[ n+1 ]^2 ]-PrimePi[ Prime[ n ]^2 ]
%Y A050216 First differences of A000879. Cf. A089609.
%K A050216 nonn,new
%O A050216 1,1
%A A050216 Eric Weisstein (eric(AT)weisstein.com)
%E A050216 Edited by njas, Nov 15 2009

%I A167906
%S A167906 1,2,3,4,14,17,18,19,20,21,22,35,36,37,42,44,45,46,47,48,49,63,64,65,66,
%T A167906 67,86,89,90,91,92,93,94,107,108,109,110,111,112,123,132,134,135,136,
%U A167906 137,146,148,161,162,168,170,171,179,180,185,186,187,189,191,192,193
%N A167906 Fixed points of permutations A121878, A167904, A167905.
%C A167906 A121878(a(n))=a(n); A167904(a(n))=a(n); A167905(a(n))=a(n).
%K A167906 nonn,new
%O A167906 1,2
%A A167906 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167907
%S A167907 3,5,7,10,11,13,15,17,19,21,23,26,29,30,29,30,35,37,39,41,43,46,47,51,
%T A167907 53,51,53,57,59,61,65,67,65,66,71,73,77,78,77,82,85,83,85,89,91,93,95,
%U A167907 97,101,102,101,105,107,109,111,113,115,118,123,122,119,122,127,129,131
%N A167907 A121878(n) + A121878(n+1).
%C A167907 A008966(a(n)) = 1 by definition of A121878.
%Y A167907 Cf. A167903.
%K A167907 nonn,new
%O A167907 1,1
%A A167907 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167905
%S A167905 1,2,3,4,6,5,8,7,10,9,12,11,16,14,13,15,17,18,19,20,21,22,24,23,26,27,
%T A167905 28,25,30,29,34,31,32,33,35,36,37,39,40,38,43,42,41,44,45,46,47,48,49,
%U A167905 51,52,50,54,53,56,55,58,57,62,61,59,60,63,64,65,66,67,69,70,68,72,71
%N A167905 Inverse integer permutation to A121878.
%C A167905 a(A167906(n)) = A167906(n);
%C A167905 a(A121878(n)) = A121878(a(n)) = n;
%C A167905 a(A167904(n)) = A167904(a(n)) = A121878(n).
%H A167905 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A167905 nonn,new
%O A167905 1,2
%A A167905 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167902
%S A167902 1,7,2,4,3,8,11,5,12,6,13,9,10,14,19,24,18,15,21,23,20,16,17,25,22,26,
%T A167902 31,27,30,33,29,28,32,34,37,39,36,35,40,38,41,47,42,49,43,48,44,50,45,
%U A167902 46,51,62,52,57,53,61,54,56,55,63,60,58,59,64,67,69,66,65,70,68,71,75
%N A167902 Inverse integer permutation to A075380.
%C A167902 a(A075381(n)) = A075381(n);
%C A167902 a(A075380(n)) = A075380(a(n)) = n;
%C A167902 a(A167901(n)) = A167901(a(n)) = A121878(n).
%H A167902 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A167902 nonn,new
%O A167902 1,2
%A A167902 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167904
%S A167904 1,2,3,4,5,6,7,8,9,10,11,12,16,14,13,15,17,18,19,20,21,22,23,24,27,28,
%T A167904 25,26,29,30,33,34,31,32,35,36,37,39,40,38,41,42,43,44,45,46,47,48,49,
%U A167904 51,52,50,53,54,55,56,57,58,60,59,62,61,63,64,65,66,67,69,70,68,71,72
%N A167904 A121878(A121878(n)).
%C A167904 Permutation of positive integers;
%C A167904 a(A167906(n)) = A167906(n);
%C A167904 a(A167905(n)) = A167905(a(n)) = A121878(n).
%H A167904 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A167904 nonn,new
%O A167904 1,2
%A A167904 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167903
%S A167903 4,8,9,12,18,12,8,18,25,20,16,20,25,32,40,45,40,32,36,40,44,45,36,40,50,
%T A167903 54,60,63,60,56,60,63,64,72,75,72,75,76,75,80,84,88,92,96,99,92,88,90,
%U A167903 92,99,104,108,112,116,117,112,116,125,124,117,108,112,124,132,135,132
%N A167903 A075380(n) + A075380(n+1).
%C A167903 A008966(a(n)) = 0 by definition of A075380.
%Y A167903 Cf. A167907.
%K A167903 nonn,new
%O A167903 1,1
%A A167903 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167901
%S A167901 1,5,8,4,6,13,3,10,9,11,2,12,7,14,17,25,20,23,18,19,15,24,21,22,16,26,
%T A167901 32,33,27,31,28,30,29,34,40,35,38,39,37,36,41,45,49,42,44,48,43,50,47,
%U A167901 46,51,55,59,54,63,62,57,52,60,56,58,53,61,64,70,65,68,69,67,66,71,74
%N A167901 A075380(A075380(n)).
%C A167901 Permutation of positive integers;
%C A167901 a(A075381(n)) = A075381(n);
%C A167901 a(A167902(n)) = A167902(a(n)) = A075380(n).
%H A167901 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%K A167901 nonn,new
%O A167901 1,2
%A A167901 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2009

%I A167895
%S A167895 3,5,7,11,29,41,47,73,107,137,167
%N A167895 Primes in A175040.
%Y A167895 Cf. A175040.
%K A167895 nonn,new
%O A167895 1,1
%A A167895 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Nov 15 2009

%I A167894
%S A167894 1,1,1,3,13,71,461,3447,29093,273343,2829325,31998903,
%T A167894 392743957,5201061455,73943424413,1123596277863,18176728317413,
%U A167894 311951144828863,5661698774848621,108355864447215063
%V A167894 1,-1,-1,-3,-13,-71,-461,-3447,-29093,-273343,-2829325,-31998903,
%W A167894 -392743957,-5201061455,-73943424413,-1123596277863,-18176728317413,
%X A167894 -311951144828863,-5661698774848621,-108355864447215063
%N A167894 O.g.f. inverse of factorial numbers A000142 o.g.f..
%Y A167894 Cf. A003319, A158882
%K A167894 sign,new
%O A167894 0,4
%A A167894 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2009

%I A167893
%S A167893 1,9,134,2878,76966,2376934,81330523
%N A167893 a(n) = Sum[ CatalanNumber[k]^3, {k, 1, n} ].
%C A167893 CatalanNumber[k] = (2k)!/k!/(k+1)! = Binomial[2k,k]/(k+1).
%C A167893 For prime p=7 p^2 divides a(p^2) and p divides all a(n) from n=(p^2-1)/2 to n=p^2-2.
%C A167893 For primes p={19,97} p divides all a(n) from n=(p-1)/2 to n=p-2.
%H A167893 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>
%F A167893 a(n) = Sum[ CatalanNumber[k]^3, {k, 1, n} ].
%F A167893 a(n) = Sum[ ((2k)!/k!/(k+1)!)^3, {k, 1, n} ].
%F A167893 a(n) = Sum[ A000108(k)^3, {k, 1, n} ].
%F A167893 a(n) = Sum[ A033536(k), {k, 1, n} ].
%Y A167893 Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.
%K A167893 more,nonn,new
%O A167893 1,2
%A A167893 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 15 2009

%I A167892
%S A167892 1,5,30,226,1990,19414,203455
%N A167892 a(n) = Sum[ CatalanNumber[k]^2, {k, 1, n} ].
%C A167892 CatalanNumber[k] = (2k)!/k!/(k+1)! = Binomial[2k,k]/(k+1).
%H A167892 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>
%F A167892 a(n) = Sum[ CatalanNumber[k]^2, {k, 1, n} ].
%F A167892 a(n) = Sum[ ((2k)!/k!/(k+1)!)^2, {k, 1, n} ].
%F A167892 a(n) = Sum[ A000108(k)^2, {k, 1, n} ].
%F A167892 a(n) = Sum[ A001246(k), {k, 1, n} ].
%F A167892 a(n) = A094639(n) - 1.
%Y A167892 Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.
%K A167892 more,nonn,new
%O A167892 1,2
%A A167892 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 15 2009

%I A167005
%S A167005 1,9,243,59076,111615732,1491228760410,142100883744107238,
%T A167005 100726960061072884260690,551358313898624152526339325936,
%U A167005 23916527760781825204570536862624479480
%N A167005 G.f.: A(x) = Sum_{n>=0} 3^n*log(G(3^n*x))^n/n! where G(x) = g.f. of A167003.
%C A167005 The g.f. of A167003, G(x), satisfies:
%C A167005 Sum_{n>=0} log(G(3^n*x))^n/n! = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
%F A167005 a(n) = [x^n] G(x)^(3^(n+1)) for n>=0 where G(x) = g.f. of A167003.
%e A167005 G.f.: A(x) = 1 + 9*x + 243*x^2 + 59076*x^3 + 111615732*x^4 +...
%e A167005 Let G(x) equal the g.f. of A167003:
%e A167005 G(x) = 1 + x - 4*x^2 - 4*x^3 - 8220*x^4 - 16910960*x^5 - 220513689396*x^6 +...
%e A167005 then the g.f. A(x) of this sequence equals the series:
%e A167005 A(x) = 1 + 3*log(G(3x)) + 9*log(G(9x))^2/2! + 27*log(G(27x))^3/3! + 81*log(G(81x))^4/4! +...
%e A167005 ILLUSTRATE (3^n)-th POWERS OF G.F. G(x) OF A167003.
%e A167005 The coefficients in the expansion of G(x)^(3^n), n>=0, begin:
%e A167005 G^1: [1, 1, -4, -4, -8220, -16910960, -220513689396,...];
%e A167005 G^3: [(1), 3, -9, -35, -24648, -50782068, -661642361248,...];
%e A167005 G^9: [1, (9), 0, -240, -74574, -152788194, -1985840486856,...];
%e A167005 G^27: [1, 27, (243), 9, -236682, -462449898, -5965789971726,...];
%e A167005 G^81: [1, 81, 2916, (59076), 0, -1420876404, -17973134801100,...];
%e A167005 G^243: [1, 243, 28431, 2125845, (111615732), 0, -54490964413644,...];
%e A167005 G^729: [1, 729, 262440, 62178840, 10895760846, (1491228760410), 0,...]; ...
%e A167005 where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.
%o A167005 (PARI) {a(n)=local(A=[1,9],B=[1,3],G=[1,1]);for(i=1,n,G=concat(G,0); B=Vec(sum(m=0,#G,log(subst(Ser(G),x,3^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/3^(#G-1))); A=Vec(sum(m=0,#G,3^m*log(subst(Ser(G),x,3^m*x))^m/m!)); A[n+1]}
%Y A167005 Cf. A167003, A167004.
%K A167005 nonn,new
%O A167005 0,2
%A A167005 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2009

%I A167004
%S A167004 1,3,0,9,0,0,0,0,0,2187,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,282429536481,
%T A167004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A167004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167004 Least possible nonnegative coefficients of x^n in G(x)^(3^n), n>=0, such that G(x) is an integer series with G'(0)=G(0)=1; the G(x) that satisfies this condition equals the g.f. of A167003.
%F A167004 G.f.: A(x) = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
%F A167004 G.f.: A(x) = Sum_{n>=0} log(G(3^n*x))^n/n! where G(x) = g.f. of A167003.
%F A167004 a(n) = [x^n] G(x)^(3^n) for n>=0 where G(x) = g.f. of A167003.
%e A167004 G.f.: A(x) = 1 + 3*x + 9*x^3 + 2187*x^9 + 282429536481*x^27 +...
%e A167004 A(x) = 1 + 3^(1-0)*x + 3^(3-1)*x^3 + 3^(9-2)*x^9 + 3^(27-3)*x^27 + 3^(81-4)*x^81 +...
%e A167004 Let G(x) equal the g.f. of A167003:
%e A167004 G(x) = = 1 + x - 4*x^2 - 4*x^3 - 8220*x^4 - 16910960*x^5 - 220513689396*x^6 +...
%e A167004 then the g.f. A(x) of this sequence equals the series:
%e A167004 A(x) = 1 + log(G(3x)) + log(G(9x))^2/2! + log(G(27x))^3/3! + log(G(81x))^4/4! +...
%e A167004 ILLUSTRATE (3^n)-th POWERS OF G.F. G(x) OF A167003.
%e A167004 The coefficients in the expansion of G(x)^(3^n), n>=0, begin:
%e A167004 G^1: [(1), 1, -4, -4, -8220, -16910960, -220513689396,...];
%e A167004 G^3: [1, (3), -9, -35, -24648, -50782068, -661642361248,...];
%e A167004 G^9: [1, 9, (0), -240, -74574, -152788194, -1985840486856,...];
%e A167004 G^27: [1, 27, 243, (9), -236682, -462449898, -5965789971726,...];
%e A167004 G^81: [1, 81, 2916, 59076, (0), -1420876404, -17973134801100,...];
%e A167004 G^243: [1, 243, 28431, 2125845, 111615732, (0), -54490964413644,...];
%e A167004 G^729: [1, 729, 262440, 62178840, 10895760846, 1491228760410, (0),...]; ...
%e A167004 where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence and equal 3^(3^m-m) at positions n=3^m for m>=0, with zeros elsewhere (except for the initial '1').
%o A167004 (PARI) {a(n)=if(n==0,1,if(n==3^valuation(n,3),3^(n-valuation(n,3)),0))}
%o A167004 (PARI) /* A(x) = Sum_{n>=0} log(G(3^n*x))^n/n!, G(x) = g.f. of A167003: */ {a(n)=local(A=[1,3],G=[1,1]);for(i=1,n,G=concat(G,0); A=Vec(sum(m=0,#G,log(subst(Ser(G),x,3^m*x))^m/m!)); G[ #G]=-floor(A[ #G]/3^(#G-1)));A[n+1]}
%Y A167004 Cf. A167003, A167005, variant: A167001.
%K A167004 nonn,new
%O A167004 0,2
%A A167004 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2009

%I A167890
%S A167890 2381167,2435011,2507119,5407113,5411372,5411421,5411470,5411519,
%T A167890 5411567,5411568,5411569,5411570,5411571,5411572,5411573,5411617,
%U A167890 5411666,5411715,5411764,5411813,5411915,5411916,5558911,5801161
%N A167890 Numbers n with property that in each of 9 bases 2..10, n has (at least one) substring '11'.
%H A167890 Zak Seidov, <a href="b167890.txt">Table of n, a(n) for n = 1..250</a>
%e A167890 2381167_10, 4428311_9, 11052557_8, 26145115_7, 123011531_6, 1102144132_5,
%e A167890 21011111233_4, 11110222100101_3, 1001000101010101101111_2
%e A167890 2435011_10, 4521177_9, 11223703_8, 26461105_7, 124105111_6, 1110410021_5,
%e A167890 21102133003_4, 11120201012121_3, 1001010010011111000011_2.
%K A167890 base,nonn,new
%O A167890 1,1
%A A167890 Zak Seidov (zakseidov(AT)yahoo.com), Nov 14 2009

%I A167002
%S A167002 1,4,20,320,21064,5030400,4056470528,10872157339648,98162974155542592,
%T A167002 3052890463194814939136,334052589949087491382968320,
%U A167002 130858881562759880830581892710400
%N A167002 G.f.: A(x) = Sum_{n>=0} 2^n*log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.
%C A167002 The g.f. of A167000, G(x), satisfies:
%C A167002 Sum_{n>=0} log(G(2^n*x))^n/n! = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).
%F A167002 a(n) = [x^n] G(x)^(2^(n+1)) for n>=0 where G(x) = g.f. of A167000.
%e A167002 G.f.: A(x) = 1 + 4*x + 20*x^2 + 320*x^3 + 21064*x^4 + 5030400*x^5 +...
%e A167002 Let G(x) equal the g.f. of A167000:
%e A167002 G(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 - 422963721*x^7 +...
%e A167002 then the g.f. A(x) of this sequence equals the series:
%e A167002 A(x) = 1 + 2*log(G(2x)) + 4*log(G(4x))^2/2! + 8*log(G(8x))^3/3! + 16*log(G(16x))^4/4! +...
%e A167002 ILLUSTRATE (2^n)-th POWERS OF G.F. G(x) OF A167000.
%e A167002 The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
%e A167002 G^1: [1,1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
%e A167002 G^2: [(1),2,-1,-2,-31,-3566,-989830,-846910236,...];
%e A167002 G^4: [1,(4),2,-8,-69,-7252,-1993858,-1697772536,...];
%e A167002 G^8: [1,8,(20),0,-198,-15088,-4045944,-3411523840,...];
%e A167002 G^16: [1,16,104,(320),4,-33344,-8341216,-6888386304,...];
%e A167002 G^32: [1,32,464,3968,(21064),0,-17646208,-14050624512,...];
%e A167002 G^64: [1,64,1952,37632,511376,(5030400),0,-29063442432,...];
%e A167002 G^128: [1,128,8000,325120,9649952,222432256,(4056470528),0,...]; ...
%e A167002 where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.
%o A167002 (PARI) {a(n)=local(A=[1,4],B=[1,2],G=[1,1]);for(i=1,n,G=concat(G,0); B=Vec(sum(m=0,#G,log(subst(Ser(G),x,2^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/2^(#G-1))); A=Vec(sum(m=0,#G,2^m*log(subst(Ser(G),x,2^m*x))^m/m!)); A[n+1]}
%Y A167002 Cf. A167000, A167001.
%K A167002 nonn,new
%O A167002 0,2
%A A167002 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2009

%I A167891
%S A167891 1,4,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A167891 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A167891 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167891 A000004 preceded by 1, 4, 2.
%C A167891 Inverse binomial transform of A028387.
%F A167891 a(0) = 1, a(1) = 4, a(2) = 2, a(n) = 0 for n > 2.
%F A167891 G.f.: 1+4*x+2*x^2.
%o A167891 (PARI) {concat([1, 4, 2], vector(100))}
%Y A167891 Cf. A000004 (zero sequence), A028387 (n+(n+1)^2), A166926 (1, 2, 4, 0, 0, 0, 0, ...), A130706 (1, 2, 0, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, 0, ...), A167858 (3, 14, 36, 36, 12, 0, 0, 0, ...), A167876 (1, 3, 4, 2, 0, 0, 0, ...).
%K A167891 easy,nonn,new
%O A167891 0,2
%A A167891 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 14 2009

%I A167847
%S A167847 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,4567
%N A167847 Straight-line primes.
%C A167847 Prime numbers with 2 digits together with the primes whose digits are in arithmetic progression. The structure of digits represent a straight line.
%C A167847 Note that in the graphic representation the points are connected by imaginary line segments (See also A135643).
%C A167847 Note that all primes with two are straight-line primes but there are no members of this sequence with three digits.
%e A167847 The number 4567 is straight-line prime:
%e A167847 . . . .
%e A167847 . . . .
%e A167847 . . . 7
%e A167847 . . 6 .
%e A167847 . 5 . .
%e A167847 4 . . .
%e A167847 . . . .
%e A167847 . . . .
%e A167847 . . . .
%e A167847 . . . .
%Y A167847 Cf. A000040, A134811, A134951, A134971, A135643, A167841, A167842, A167843, A167844, A167845, A167846, A167853.
%K A167847 base,more,nonn,new
%O A167847 1,1
%A A167847 Omar E. Pol (info(AT)polprimos.com), Nov 14 2009

%I A167885
%S A167885 0,4,18,60,126,184,270,370,462,611,742,900,1072,1224,1501,1660,1869,
%T A167885 2244,2592,2825,3302,3537,3864,4350,4867,5216,5511,5882,6300,6912,7524,
%U A167885 8229,8920,9348,10019,10560,11295,11822,12624,13230,13850,14382,15236
%N A167885 nth single or isolated number*nth non-single or nonisolated number.
%F A167885 a(n)=A167706(n)*A167707(n).
%e A167885 a(1)=2*0=0, a(2)=4*1=4, a(3)=6*3=18, a(4)=12*5=60.
%Y A167885 Cf. A167706, A167707.
%K A167885 nonn,new
%O A167885 1,2
%A A167885 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 14 2009, Nov 15 2009

%I A167886
%S A167886 2,3,3,7,11,15,21,27,31,34,39,45,49,55,60,63,68,80,84,88,101,104,110,
%T A167886 121,126,131,134,139,145,156,160,172,183,187,190,196,206,211,215,221,
%U A167886 227,231,241,253,257,261,274,279,289,292,297,304,309,314,317,321,328
%N A167886 nth single or isolated number minus nth non-single or nonisolated number.
%F A167886 a(n)=A167706(n)-A167707(n).
%e A167886 a(1)=2-0=2, a(2)=4-1=3, a(3)=6-3=3, a(4)=12-5=7.
%Y A167886 Cf. A167706, A167707.
%K A167886 nonn,new
%O A167886 1,1
%A A167886 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 14 2009

%I A167887
%S A167887 2,5,9,17,25,31,39,47,53,60,67,75,83,89,98,103,110,124,132,138,153,158,
%T A167887 164,179,188,195,200,207,215,228,236,250,263,269,276,284,296,303,311,
%U A167887 319,327,333,345,361,367,373,388,395,407,414,421,430,437,444,449,457
%N A167887 nth single or isolated number plus nth non-single or nonisolated number.
%F A167887 a(n)=A167706(n)+A167707(n).
%e A167887 a(1)=2+0=2, a(2)=4+1=5, a(3)=6+3=9, a(4)=12+5=17.
%Y A167887 Cf. A167706, A167707.
%K A167887 nonn,new
%O A167887 1,1
%A A167887 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 14 2009

%I A167888
%S A167888 1,4,216,248832,61222032,78310985281,19683000000000
%N A167888 nth single or isolated number^nth non-single or nonisolated number.
%F A167888 a(n)=A167706(n)^A167707(n).
%e A167888 a(1)=2^0=1, a(2)=4^1=4, a(3)=6^3=216, a(4)=12^5=248832.
%Y A167888 Cf. A167706, A167707.
%K A167888 nonn,new
%O A167888 1,2
%A A167888 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 14 2009

%I A167889
%S A167889 0,0,1,4,25,100,481,1924,8425
%N A167889 a(n)=4a(n-1)+9a(n-2)-36a(n-3); a(0)=a(1)=0,a(2)=1.
%C A167889 Principal sequence.See A138587.
%F A167889 a(n+1)-4a(n)=0,1,0,9,0,81,0,729,=mix A000004,A001019.
%K A167889 nonn,uned,new
%O A167889 0,4
%A A167889 Paul Curtz (bpcrtz(AT)free.fr), Nov 14 2009

%I A167001
%S A167001 1,2,2,0,4,0,0,0,32,0,0,0,0,0,0,0,4096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A167001 134217728,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A167001 0,288230376151711744,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167001 Least possible nonnegative coefficients of x^n in G(x)^(2^n), n>=0, for an integer series G(x) such that G'(0)=G(0)=1; the G(x) that satisfies this condition equals the g.f. of A167000.
%F A167001 G.f.: A(x) = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).
%F A167001 G.f.: A(x) = Sum_{n>=0} log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.
%F A167001 a(n) = [x^n] G(x)^(2^n) for n>=0 where G(x) = g.f. of A167000.
%e A167001 G.f.: A(x) = 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...
%e A167001 A(x) = 1 + 2^(1-0)*x + 2^(2-1)*x^2 + 2^(4-2)*x^4 + 2^(8-3)*x^8 + 2^(16-4)*x^16 +...
%e A167001 Let G(x) equal the g.f. of A167000:
%e A167001 G(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 - 422963721*x^7 +...
%e A167001 then the g.f. A(x) of this sequence equals the series:
%e A167001 A(x) = 1 + log(G(2x)) + log(G(4x))^2/2! + log(G(8x))^3/3! + log(G(16x))^4/4! +...
%e A167001 ILLUSTRATE (2^n)-th POWERS OF G.F. G(x) OF A167000.
%e A167001 The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
%e A167001 G^1: [(1),1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
%e A167001 G^2: [1,(2),-1,-2,-31,-3566,-989830,-846910236,...];
%e A167001 G^4: [1,4,(2),-8,-69,-7252,-1993858,-1697772536,...];
%e A167001 G^8: [1,8,20,(0),-198,-15088,-4045944,-3411523840,...];
%e A167001 G^16: [1,16,104,320,(4),-33344,-8341216,-6888386304,...];
%e A167001 G^32: [1,32,464,3968,21064,(0),-17646208,-14050624512,...];
%e A167001 G^64: [1,64,1952,37632,511376,5030400,(0),-29063442432,...];
%e A167001 G^128: [1,128,8000,325120,9649952,222432256,4056470528,(0),...]; ...
%e A167001 where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence and equal 2^(2^m-m) at positions n=2^m for m>=0, with zeros elsewhere (except for the initial '1').
%o A167001 (PARI) {a(n)=if(n==0,1,if(n==2^valuation(n,2),2^(n-valuation(n,2)),0))}
%o A167001 (PARI) /* A(x) = Sum_{n>=0} log(G(2^n*x))^n/n!, G(x) = g.f. of A167000: */ {a(n)=local(A=[1,2],G=[1,1]);for(i=1,n,G=concat(G,0); A=Vec(sum(m=0,#G,log(subst(Ser(G),x,2^m*x))^m/m!)); G[ #G]=-floor(A[ #G]/2^(#G-1)));A[n+1]}
%Y A167001 Cf. A167000, A167002, variant: A167004.
%K A167001 nonn,new
%O A167001 0,2
%A A167001 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2009

%I A167884
%S A167884 1,1,1,1,18,1,1,179,179,1,1,1636,6086,1636,1,1,14757,144362,144362,
%T A167884 14757,1,1,132854,2941135,7218100,2941135,132854,1,1,1195735,55446309,
%U A167884 277509955,277509955,55446309,1195735,1,1,10761672,1001178268
%N A167884 Triangle of recursion:m=8;A(n,k) := (m*n - m*k + 1)A(n - 1, k - 1) + (m*k - (m - 1))A(n - 1, k)
%C A167884 Row sums are:
%C A167884 {1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000,...}
%C A167884 The importance of this recursion is that it gives an integer inverse z transform polynomial set:
%C A167884 p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n;
%C A167884 b = Table[p[x, n], {n, 0, 10}];
%C A167884 Table[CoefficientList[ExpandAll[InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]
%e A167884 {1},
%e A167884 {1, 1},
%e A167884 {1, 18, 1},
%e A167884 {1, 179, 179, 1},
%e A167884 {1, 1636, 6086, 1636, 1},
%e A167884 {1, 14757, 144362, 144362, 14757, 1},
%e A167884 {1, 132854, 2941135, 7218100, 2941135, 132854, 1},
%e A167884 {1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1},
%e A167884 {1, 10761672, 1001178268, 9211047544, 18315657030, 9211047544, 1001178268, 10761672, 1},
%e A167884 {1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1}
%t A167884 Clear[A, p, n, k]
%t A167884 m = 8
%t A167884 A[n_, 1] := 1
%t A167884 A[n_, n_] := 1
%t A167884 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
%t A167884 a = Table[A[n, k], {n, 10}, {k, n}]
%t A167884 Flatten[a]
%K A167884 nonn,uned,new
%O A167884 1,5
%A A167884 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 14 2009

%I A167883
%S A167883 1,1,2,1,10,10,1,32,90,60,1,74,594,1040,520,1,224,2226,6684,7800,3120,1,
%T A167883 352,12124,95304,227052,215280,71760,1,1058,38484,358656,1252980,
%U A167883 2008152,1506960,430560,1,1348,142264,4028712,32909556,97352640
%N A167883 Coefficients of a recursive polynomial:p(k,n)=If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))] ( correction with cubic term in the infinite sum)
%C A167883 Row sums are:
%C A167883 {1, 3, 21, 183, 2229, 20055, 621873, 5596851, 374989401, 3374904603, 422613054909,...}
%C A167883 The remarkable thing about these polynomials is that there infinite sums are a symmetrical triangle.
%C A167883 Quadratic {1,18,1} type approximates the general Pascal recursion which is the first,
%C A167883 since the MacMahon not to be a rational inverse z Transform.
%F A167883 p(k,0)=1;
%F A167883 p(k,1)=1+2*k; p(k,n)=If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))]
%e A167883 {1},
%e A167883 {1, 2},
%e A167883 {1, 10, 10},
%e A167883 {1, 32, 90, 60},
%e A167883 {1, 74, 594, 1040, 520},
%e A167883 {1, 224, 2226, 6684, 7800, 3120},
%e A167883 {1, 352, 12124, 95304, 227052, 215280, 71760},
%e A167883 {1, 1058, 38484, 358656, 1252980, 2008152, 1506960, 430560},
%e A167883 {1, 1348, 142264, 4028712, 32909556, 97352640, 132914880, 86112000, 21528000},
%e A167883 {1, 4046, 434880, 12939720, 122900940, 489515256, 982860480, 1055825280, 581256000, 129168000},
%e A167883 {1, 4598, 1184922, 92796080, 2442817180, 21051364536, 73606098792, 129668682240, 123157690560, 60493680000, 12098736000}
%e A167883 The infinite sum triangle is:
%e A167883 Table[Sum[p[k, n]*x^k, {k, 0, Infinity}], {n, 0, 10}];
%e A167883 {1},
%e A167883 {1, 1},
%e A167883 {-1, -18, -1},
%e A167883 {1, 179, 179, 1}
%e A167883 {-1, -2224, -8030, -2224, -1},
%e A167883 {1, 20049, 167150, 167150, 20049, 1},
%e A167883 {-1, -5596844, -145462469, -524422080, -360876091, -48653716, 1},
%e A167883 {1, 5596843, 194116185, 885298171, 885298171, 194116185, 5596843, 1},
%e A167883 {-1, -374989392, -25339790572, -207966886768, -400645626534, -207966886768, -25339790572, -374989392, -1}
%t A167883 Clear[p, x, n, k, a]
%t A167883 p[k, 0] := 1; p[k, 1] := 1 + 2*k;
%t A167883 p[k_, n_] := If[Mod[n, 2] == 0, (1 + 2*k)*p[k, n - 1] + n*Binomial[n + 1, n - 1]*k*(k + 1)*p[k, n - 2], (1 + 2*k)*(1 + 3*(p[k, n - 1] - 1))];
%t A167883 Table[CoefficientList[ExpandAll[p[k, n]], k], {n, 0, 10}];
%t A167883 Flatten[%]
%K A167883 nonn,uned,new
%O A167883 0,3
%A A167883 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 14 2009

%I A167877
%S A167877 0,1,0,3,4,3,2,1,0,9,10,9,12,13,12,11,10,9,8,7,6,5,4,3,2,1,0,27,28,27,
%T A167877 30,31,30,29,28,27,36,37,36,39,40,39,38,37,36,35,34,33,32,31,30,29,28,
%U A167877 27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2
%N A167877 Largest m<=n such that no carry occurs when adding m to n in ternary arithmetic.
%C A167877 A167878(n) = a(n) + n.
%H A167877 R. Zumkeller, <a href="b167877.txt">Table of n, a(n) for n = 0..10000</a>
%Y A167877 Cf. A007089, A167831, A035327 for the decimal and binary cases.
%K A167877 base,nonn,new
%O A167877 0,4
%A A167877 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009

%I A167832
%S A167832 0,2,4,6,8,9,9,9,9,9,20,22,24,26,28,29,29,29,29,29,40,42,44,46,48,49,49,
%T A167832 49,49,49,60,62,64,66,68,69,69,69,69,69,80,82,84,86,88,89,89,89,89,89,
%U A167832 99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99,99
%N A167832 A167831(n) + n.
%C A167832 No carry occurs when calculating a(n) by adding A167831(n) to n in decimal arithmetic.
%H A167832 R. Zumkeller, <a href="b167832.txt">Table of n, a(n) for n = 0..9999</a>
%Y A167832 Cf. A167878, A003817 for the ternary and binary cases.
%K A167832 base,nonn,new
%O A167832 0,2
%A A167832 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009

%I A167878
%S A167878 0,2,2,6,8,8,8,8,8,18,20,20,24,26,26,26,26,26,26,26,26,26,26,26,26,26,
%T A167878 26,54,56,56,60,62,62,62,62,62,72,74,74,78,80,80,80,80,80,80,80,80,80,
%U A167878 80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80
%N A167878 A167877(n) + n.
%C A167878 No carry occurs when calculating a(n) by adding A167877(n) to n in ternary arithmetic.
%H A167878 R. Zumkeller, <a href="b167878.txt">Table of n, a(n) for n = 0..10000</a>
%Y A167878 Cf. A007089, see A167832, A003817 for the decimal and binary cases.
%K A167878 base,nonn,new
%O A167878 0,2
%A A167878 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009

%I A167831
%S A167831 0,1,2,3,4,4,3,2,1,0,10,11,12,13,14,14,13,12,11,10,20,21,22,23,24,24,23,
%T A167831 22,21,20,30,31,32,33,34,34,33,32,31,30,40,41,42,43,44,44,43,42,41,40,
%U A167831 49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27
%N A167831 Largest m<=n such that no carry occurs when adding m to n in decimal arithmetic.
%C A167831 A167832(n) = a(n) + n.
%H A167831 R. Zumkeller, <a href="b167831.txt">Table of n, a(n) for n = 0..9999</a>
%Y A167831 Cf. A167877, A035327 for the ternary and binary cases.
%K A167831 base,nonn,new
%O A167831 0,3
%A A167831 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009

%I A167876
%S A167876 1,3,4,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A167876 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A167876 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167876 A000004 preceded by 1, 3, 4, 2.
%C A167876 Inverse binomial transform of A167875.
%F A167876 a(0) = 1, a(1) = 3, a(2) = 4, a(3) = 2, a(n) = 0 for n > 3.
%F A167876 G.f.: (1+x)*(1+2*x+2*x^2).
%o A167876 (PARI) {concat([1, 3, 4, 2], vector(99))}
%Y A167876 Cf. A000004 (zero sequence), A167875 (one third of product plus sum of three consecutive nonnegative integers), A166926 (1, 2, 4, 0, 0, 0, 0, ...), A130706 (1, 2, 0, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, 0, ...), A167858 (3, 14, 36, 36, 12, 0, 0, 0, ...).
%K A167876 easy,nonn,new
%O A167876 0,2
%A A167876 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 14 2009

%I A167875
%S A167875 1,4,11,24,45,76,119,176,249,340,451,584,741,924,1135,1376,1649,1956,
%T A167875 2299,2680,3101,3564,4071,4624,5225,5876,6579,7336,8149,9020,9951,10944,
%U A167875 12001,13124,14315,15576,16909,18316,19799,21360,23001,24724,26531
%N A167875 One third of product plus sum of three consecutive nonnegative integers.
%C A167875 a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.
%C A167875 Equals A006527 without initial term 0: a(n) = A006527(n+1).
%C A167875 Binomial transform of A167876.
%C A167875 Inverse binomial transform of A080930.
%C A167875 a(n) = A007290(n+2)+n+1.
%C A167875 a(n) = A014820(n)/(n+1) for n > 0.
%C A167875 a(n) = A116731(n+2)-1.
%C A167875 a(n) = A033547(n+1)-n.
%C A167875 a(n) = A054602(n)/3.
%C A167875 a(n) = A086514(n+3)-2.
%C A167875 a(n) = A002061(n+1)+a(n-1) for n > 0.
%C A167875 a(n) = A005894(n)-a(n-1) for n > 0.
%C A167875 First bisection is A057813.
%C A167875 Second differences are in A004277.
%F A167875 a(n) = (n^3+3*n^2+5*n+3)/3 for n > 0.
%F A167875 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2 for n > 3; a(0)=1, a(1)=4, a(2)=11, a(3)=24.
%F A167875 G.f.: (1+x^2)/(1-x)^4.
%e A167875 a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
%e A167875 a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
%o A167875 (MAGMA) [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];
%Y A167875 Cf. A001477 (nonnegative integers), A006527 ((n^3+2*n)/3), A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...), A080930, A007290 (2*C(n, 3)), A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2), A116731, A033547 (n*(n^2+5)/3), A054602 (Sum_{d|3} phi(d)*n^(3/d)),
%Y A167875 A086514 ((n^3-6*n^2+14*n-6)/3), A002061 (n^2-n+1), A005894 (centered tetrahedral numbers), A057813 ((2*n+1)*(4*n^2+4*n+3)/3), A004277 (1 and the positive even numbers), A028387 (n+(n+1)^2), A166941, A166942, A166943.
%K A167875 nonn,new
%O A167875 0,2
%A A167875 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 14 2009

%I A167874
%S A167874 3,15,69,409,2447,16337,117959,880623,6849011,54825357,448752095,
%T A167874 3741170439,31669329743,271560643329,2354418484607,20608371394595,
%U A167874 181897678706317,1617351777154871
%N A167874 The number of distinct primes < 10^n that enter in the composition of twin-prime pairs.
%C A167874 Number of terms in A001097 with at most n digits.
%F A167874 a(n) = 2*A007508(n) - 1. The entries are odd because of 5 that belongs to both of the first two twin-prime pairs (3,5) and (5,7).
%e A167874 There are exactly a(2) = 15 distinct primes below 10^2 that form twin primes (p, p+2), namely, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73 : (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
%Y A167874 Cf. A001097, A077800, A007508
%K A167874 nonn,uned,new
%O A167874 1,1
%A A167874 Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 14 2009

%I A167873
%S A167873 10,6,10,4,10,6,10,4,10,6,10,4,10,6,10,4,10,6,10,4,10,6,10,4
%N A167873 Periodic sequence (10, 6, 10, 4)
%C A167873 Number of different remainders mod 10 of n-th powers (i.e. number of possible last decimal digits of n-th powers).
%e A167873 a(4)=4 because the possible last decimal digits of 4-th powers are: 0, 1, 5, 6.
%K A167873 easy,nonn,new
%O A167873 1,1
%A A167873 Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Nov 14 2009

%I A167872
%S A167872 1,3,21,207,2529,36243,591381,10786527,217179009
%N A167872 a sequence of moments in relation with Feynman numbers ( A00698)
%C A167872 a(n) is the moment of order 2*n for the probability density function definite by rho(x)=sqrt(Pi/2)*exp(-x^2/2)/((xphi(x)+1)^2+Pi^2*x^2*exp(-x^2)), with phi(x)=int(t*ln(abs(x-t))*exp(-t^2/2),t=-infinity..infinity)
%D A167872 Roland Groux. Polynsomes orthogonaux et transformations integrales. Cepadues. 2008 pages 195..206
%F A167872 sum(a(n)/z^(2n+1),n=0..infinity)=(1/2)*(z-S(z)/(z*S(z)-1) with S(z)=sum((2*n)!/(2^n*n!*z^(2*n+1)), n=0..infinity
%Y A167872 a(0)=1 , for n >0 a(n)=A000698(n+2)/2-sum(A000698(n+1-k)*a(k), k=0..n-1)
%K A167872 nonn,new
%O A167872 0,2
%A A167872 Roland Groux (roland.groux(AT)orange.fr), Nov 14 2009

%I A167003
%S A167003 1,1,4,4,8220,16910960,220513689396,19336259194582782,
%T A167003 12353453824556774353132,60817754630605750994570243653,
%U A167003 2385117541132316928253481462547625452
%V A167003 1,1,-4,-4,-8220,-16910960,-220513689396,-19336259194582782,
%W A167003 -12353453824556774353132,-60817754630605750994570243653,
%X A167003 -2385117541132316928253481462547625452
%N A167003 G.f. A(x) satisfies: Sum_{n>=0} log(A(3^n*x))^n/n! = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
%F A167003 The coefficient of x^(3^n) in A(x)^(3^(3^n)) equals 3^(3^n-n):
%F A167003 [x^(3^n)] A(x)^(3^(3^n)) = 3^(3^n-n); while
%F A167003 [x^n] A(x)^(3^n) = 0 when n>0 is not a power of 3, with A(0)=1.
%e A167003 G.f.: A(x) = 1 + x - 4*x^2 - 4*x^3 - 8220*x^4 - 16910960*x^5 +...
%e A167003 log(A(x)) = x - 9*x^2/2 + x^3/3 - 32913*x^4/4 - 84513699*x^5/5 +...
%e A167003 ILLUSTRATE THE SERIES DEFINITION:
%e A167003 1 + log(A(3x)) + log(A(9x))^2/2! + log(A(27x))^3/3! + log(A(81x))^4/4! +...
%e A167003 = 1 + 3*x + 9*x^3 + 2187*x^9 + 282429536481*x^27 +...
%e A167003 = 1 + 3^(1-0)*x + 3^(3-1)*x^3 + 3^(9-2)*x^9 + 3^(27-3)*x^27 +...
%e A167003 ILLUSTRATE (3^n)-th POWERS OF G.F. A(x).
%e A167003 The coefficients in the expansion of A(x)^(3^n) for n>=0 begin:
%e A167003 n=0: [(1), 1, -4, -4, -8220, -16910960, -220513689396,...];
%e A167003 n=1: [1, (3), -9, -35, -24648, -50782068, -661642361248,...];
%e A167003 n=2: [1, 9, (0), -240, -74574, -152788194, -1985840486856,...];
%e A167003 n=3: [1, 27, 243, (9), -236682, -462449898, -5965789971726,...];
%e A167003 n=4: [1, 81, 2916, 59076, (0), -1420876404, -17973134801100,...];
%e A167003 n=5: [1, 243, 28431, 2125845, 111615732, (0), -54490964413644,...];
%e A167003 n=6: [1, 729, 262440, 62178840, 10895760846, 1491228760410, (0),...];
%e A167003 where the coefficients along the diagonal (in parenthesis) begin:
%e A167003 [1,3,0,9,0,0,0,0,0,2187,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 282429536481,...]
%e A167003 and equal 3^(3^m-m) at positions n=3^m for m>=0, with zeros elsewhere (except for the initial '1').
%o A167003 (PARI) {a(n)=local(A=[1,1],B=[1,3]);for(i=1,n,A=concat(A,0); B=Vec(sum(m=0,#A,log(subst(Ser(A),x,3^m*x))^m/m!)); A[ #A]=-floor(B[ #A]/3^(#A-1)));A[n+1]}
%Y A167003 Cf. A167004, A167005, variant: A167000.
%K A167003 sign,new
%O A167003 0,3
%A A167003 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2009

%I A167871
%S A167871 1,72,4824,316736,20614104
%N A167871 a(n) = 64^n*Sum_{ k=0..n } binomial(2*k,k)^3/64^k
%C A167871 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C A167871 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%C A167871 p^2 divides all a(n) from n = (p-1)/2 to n = p-1 for prime p of the form p = 4k+3, p = {3,7,11,19,23,31,43,47,59,...} = A002145.
%C A167871 p^2 divides all a(n) from n = (2p-1 - (p-1)/2) to n = 2p-1 for prime p of the form p = 4k+3.
%C A167871 p^2 divides all a(n) from n = (3p-1 - (p-1)/2) to n = 3p-1 for prime p of the form p = 4k+3.
%C A167871 p^2 divides all a(n) from n = (p^2-1)/2 to n = p^2-1 for prime p of the form p = 4k+3.
%F A167871 a(n) = 64^n*Sum[ Binomial[2*k,k]^3/64^k, {k,0,n} ].
%Y A167871 Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y A167871 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859, A002145.
%K A167871 more,nonn,new
%O A167871 0,2
%A A167871 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 14 2009

%I A167870
%S A167870 1,24,600,17600,624600,25996608,1204834752
%N A167870 a(n) = 16^n*Sum_{ k=0..n } binomial(2*k,k)^3/16^k
%C A167870 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C A167870 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%F A167870 a(n) = 16^n*Sum[ Binomial[2*k,k]^3/16^k, {k,0,n} ].
%Y A167870 Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y A167870 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.
%K A167870 more,nonn,new
%O A167870 0,2
%A A167870 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 14 2009

%I A167867
%S A167867 1,10,236,8472,359944,16722896
%N A167867 a(n) = 2^n*Sum_{ k=0..n } binomial(2*k,k)^3/2^k
%C A167867 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C A167867 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%F A167867 a(n) = 2^n*Sum[ Binomial[2*k,k]^3/2^k, {k,0,n} ].
%Y A167867 Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y A167867 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.
%K A167867 more,nonn,new
%O A167867 0,2
%A A167867 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 14 2009

%I A167868
%S A167868 1,11,249,8747,369241,17110731
%N A167868 a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)^3/3^k
%C A167868 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C A167868 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%F A167868 a(n) = 3^n*Sum[ Binomial[2*k,k]^3/3^k, {k,0,n} ].
%Y A167868 Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y A167868 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.
%K A167868 more,nonn,new
%O A167868 0,2
%A A167868 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 14 2009

%I A167869
%S A167869 1,12,264,9056,379224,17519904
%N A167869 a(n) = 4^n*Sum_{ k=0..n } binomial(2*k,k)^3/4^k
%C A167869 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C A167869 The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%F A167869 a(n) = 4^n*Sum[ Binomial[2*k,k]^3/4^k, {k,0,n} ].
%Y A167869 Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y A167869 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.
%K A167869 more,nonn,new
%O A167869 0,2
%A A167869 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 14 2009

%I A167866
%S A167866 0,0,1,1,1,1,2,1,2,2,2,1,2,1,3,2,2,1,3,1,3,3,3,1,2,2,3,3,3,1,4,1,3,3,3,
%T A167866 3,3,1,4,3,3,1,4,1,4,4,4,1,3,3,3,3,3,1,4,3,4,4,4,1,4,1,5,4,3,3,4,1,4,4,
%U A167866 4,1,4,1,4,4,4,3,5,1,4,4,4,1,4,3,5,4,4,1,5,3,5,5,5,4,3,1,4,4,4,1,4,1,4
%N A167866 Length of the longest partition of n into distinct parts greater than 1, with each part divisible by the next one.
%C A167866 Formally speaking, a(1) is not defined but letting a(1)=0 does not break any formula.
%F A167866 a(0)=a(1)=0 and for n>=2, a(n) = 1 + \max_{d|n, d>1} a((n-d)/d)
%o A167866 (PARI) { A167866(n) = local(r=0); if(n<=1,return(0)); fordiv(n,d, if(d>1, r=max(r,A167866((n-d)\d)); ); ); r+1 }
%Y A167866 Cf. A122651, A167439, A167865
%K A167866 nonn,new
%O A167866 0,7
%A A167866 Max Alekseyev (maxale(AT)gmail.com), Nov 13 2009

%I A167865
%S A167865 1,0,1,1,1,1,2,1,2,2,2,1,4,1,3,3,3,1,5,1,5,4,3,1,6,2,5,4,5,1,9,1,6,4,4,
%T A167865 4,8,1,6,6,7,1,11,1,8,8,4,1,10,3,10,5,8,1,11,4,10,7,6,1,13,1,10,11,7,6,
%U A167865 15,1,9,5,11,1,14,1,9,12,8,5,15,1,16,9,8,1,18,5,12,7,10,1,21,7,13,11,5
%N A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next.
%F A167865 a(0)=1 and for n>=1, a(n) = \sum_{d|n, d>1} a((n-d)/d)
%o A167865 (PARI) { A167865(n) = if(n==0,return(1)); sumdiv(n,d, if(d>1, A167865((n-d)\d) ) ) }
%Y A167865 Cf. A122651, A167439, A167866
%K A167865 nonn,new
%O A167865 0,7
%A A167865 Max Alekseyev (maxale(AT)gmail.com), Nov 13 2009

%I A167000
%S A167000 1,1,1,0,16,1767,493164,422963721,1130568823448,9811523398109059,
%T A167000 287512372919585565730,29365896347484186250530846,
%U A167000 10704256920972727382240940549099
%V A167000 1,1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,-9811523398109059,
%W A167000 -287512372919585565730,-29365896347484186250530846,
%X A167000 -10704256920972727382240940549099
%N A167000 G.f. A(x) satisfies: Sum_{n>=0} log(A(2^n*x))^n/n! = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).
%F A167000 The coefficient of x^(2^n) in A(x)^(2^(2^n)) equals 2^(2^n-n):
%F A167000 [x^(2^n)] A(x)^(2^(2^n)) = 2^(2^n-n); while
%F A167000 [x^n] A(x)^(2^n) = 0 when n>0 is not a power of 2, with A(0)=1.
%e A167000 G.f.: A(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 -...
%e A167000 log(A(x)) = x - 3*x^2/2 + 4*x^3/3 - 71*x^4/4 - 8744*x^5/5 - 2948592*x^5/5 -...
%e A167000 ILLUSTRATE THE SERIES DEFINITION:
%e A167000 1 + log(A(2x)) + log(A(4x))^2/2! + log(A(8x))^3/3! + log(A(16x))^4/4! +...
%e A167000 = 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...
%e A167000 = 1 + 2^(1-0)*x + 2^(2-1)*x^2 + 2^(4-2)*x^4 + 2^(8-3)*x^8 + 2^(16-4)*x^16 +...
%e A167000 ILLUSTRATE (2^n)-th POWERS OF G.F. A(x).
%e A167000 The coefficients in the expansion of A(x)^(2^n) for n>=0 begin:
%e A167000 [(1),1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
%e A167000 [1,(2),-1,-2,-31,-3566,-989830,-846910236,-2261982587754,...];
%e A167000 [1,4,(2),-8,-69,-7252,-1993858,-1697772536,-4527350821567,...];
%e A167000 [1,8,20,(0),-198,-15088,-4045944,-3411523840,-9068291678061,...];
%e A167000 [1,16,104,320,(4),-33344,-8341216,-6888386304,-18191329536118,...];
%e A167000 [1,32,464,3968,21064,(0),-17646208,-14050624512,-36604843747036];
%e A167000 [1,64,1952,37632,511376,5030400,(0),-29063442432,-74124859451768];
%e A167000 [1,128,8000,325120,9649952,222432256,4056470528,(0),...];
%e A167000 [1,256,32384,2698240,166530624,8117172224,325157844992,10872157339648, (32),...]; ...
%e A167000 where the coefficients along the diagonal (in parenthesis) begin:
%e A167000 [1,2,2,0,4,0,0,0,32,0,0,0,0,0,0,0,4096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 134217728,...]
%e A167000 and equal 2^(2^m-m) at positions n=2^m for m>=0, with zeros elsewhere (except for the initial '1').
%o A167000 (PARI) {a(n)=local(A=[1,1],B=[1,2]);for(i=1,n,A=concat(A,0); B=Vec(sum(m=0,#A,log(subst(Ser(A),x,2^m*x))^m/m!)); A[ #A]=-floor(B[ #A]/2^(#A-1)));A[n+1]}
%Y A167000 Cf. A167001, A167002, A167003.
%K A167000 nice,sign,new
%O A167000 0,5
%A A167000 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 13 2009
%E A167000 Typos in examples fixed by Paul D. Hanna (pauldhanna(AT)juno.com), Nov 15 2009

%I A167439
%S A167439 0,1,1,2,2,2,2,3,2,3,3,3,2,3,3,4,3,3,3,4,3,4,4,4,2,3,3,4,4,4,4,5,3,4,4,
%T A167439 4,4,4,4,5,4,4,4,5,4,5,5,5,3,4,4,4,4,4,4,5,4,5,5,5,4,5,5,6,5,4,4,5,4,5,
%U A167439 5,5,4,5,4,5,5,5,5,6,4,5,5,5,4,5,5,6,5,5,5,6,5,6,6,6,5,4,4,5,5,5,4,5,4
%N A167439 Length of the longest partition of n into distinct parts, with each part divisible by the next one.
%C A167439 a(n) > sqrt(ln(n))/2
%D A167439 V. A. Sadovnichiy, A. A. Grigoryan and S. V. Konyagin (1987), "Problems of mathematical olympiads for university students". Section 4.1, problem 25. (in Russian)
%F A167439 a(n) = max{ A167866(n), A167866(n-1)+1 }
%o A167439 (PARI) { a(n,m=0) = local(r=0); if(n==0,return(0)); fordiv(n,d, if(d<=m,next); r=max(r,1+a((n-d)\d,1)) ); r }
%Y A167439 Cf. A122651, A167439, A167865, A167866
%K A167439 nonn,new
%O A167439 0,4
%A A167439 Max Alekseyev (maxale(AT)gmail.com), Nov 13 2009, Nov 15 2009

%I A167864
%S A167864 1,5,1,4,7,7
%N A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2)))
%C A167864 Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
%D A167864 A. Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. 18 (1954) 83-87.
%D A167864 H. Delange, Sur des formules de Atle Selberg, Acta Arith. 19 (1971) 105-146.
%D A167864 G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.
%e A167864 Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.51477...
%K A167864 cons,more,nonn,new
%O A167864 1,2
%A A167864 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 13 2009

%I A167860
%S A167860 7,47,191,383,439,1151,1399,2351,2879,3119,3511,3559
%N A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).
%C A167860 Apparently A167860 is a subset of primes of the form 8n+7 (A007522).
%C A167860 A167859(n) = 4^n*Sum_{ k=0..n } ((binomial(2*k,k))^2)/4^k.
%C A167860 Every A167859(m) from m=(p-1)/2 to m=(p-1) is divisible by prime p belonging to A167860.
%C A167860 7^3 divides A167859(13) and 7^2 divides A167859(10)-A167859(13).
%C A167860 Every A167859(m) from m=(kp-1 - (p-1)/2) to m=(kp-1) is divisible by prime p from A167860.
%C A167860 Every A167859(m) from m=((p^2-1)/2) to m=(p^2-1) is divisible by prime p from A167860. For p=7 every A167859(m) from m=((p^3-1)/2) to m=(p^3-1) and from m=((p^4-1)/2) to m(p^4-1)is divisible by p^2.
%Y A167860 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859, A007522.
%K A167860 more,nonn,new
%O A167860 1,1
%A A167860 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 13 2009

%I A167859
%S A167859 1,8,68,672,7588,93856,1229200
%N A167859 a(n) = 4^n*Sum_{ k=0..n } ((binomial(2*k,k))^2)/4^k.
%C A167859 Every a(n) from a((p-1)/2) to a(p-1) is divisible by prime p for p = {7,47,191,383,439,1151,1399,2351,2879,3119,3511,3559,...} = A167860, apparently a subset of primes of the form 8n+7 (A007522).
%C A167859 7^3 divides a(13) and 7^2 divides a(10)-a(13).
%C A167859 Every a(n) from a(kp-1 - (p-1)/2) to a(kp-1) is divisible by prime p from A167860.
%C A167859 Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p from A167860. For p=7 every a(n) from a((p^3-1)/2) to a(p^3-1) and from a((p^4-1)/2) to a(p^4-1)is divisible by p^2.
%Y A167859 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167860, A007522.
%K A167859 more,nonn,new
%O A167859 0,2
%A A167859 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 13 2009

%I A167851
%S A167851 1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,
%T A167851 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,0,1,0,
%U A167851 1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,0,1,0,1,1,0,1,1,0,1
%N A167851 Triangle read by rows in which each row contains a finite triangle as shown below.
%C A167851 See also A159800, A164002, A167850 and A167852.
%e A167851 The main triangle begins:
%e A167851 1,
%e A167851 1,1,1,
%e A167851 1,1,1,1,1,1,
%e A167851 1,1,1,1,0,1,1,1,1,1,
%e A167851 1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,
%e A167851 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1,1,1,1,
%e A167851 1,1,1,1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0,1,1,1,1,1,1,1,1,
%e A167851 1,1,1,1,0,1,1,0,0,1,1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,
%e A167851 In which row 1 contains a vertex, only:
%e A167851 1,
%e A167851 Row 2 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 Row 3 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,1,1,
%e A167851 Row 4 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,1,1,1,
%e A167851 Row 5 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,1,1,1,1,
%e A167851 Row 6 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,0,0,0,1,
%e A167851 1,1,1,1,1,1,
%e A167851 Row 7 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,0,1,0,1,
%e A167851 1,0,0,0,0,1,
%e A167851 1,1,1,1,1,1,1,
%e A167851 Row 8 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,0,1,0,1,
%e A167851 1,0,1,1,0,1,
%e A167851 1,0,0,0,0,0,1,
%e A167851 1,1,1,1,1,1,1,1,
%e A167851 Row 9 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,0,1,0,1,
%e A167851 1,0,1,1,0,1,
%e A167851 1,0,1,1,1,0,1,
%e A167851 1,0,0,0,0,0,0,1,
%e A167851 1,1,1,1,1,1,1,1,1,
%e A167851 Row 10 contains the triangle:
%e A167851 1,
%e A167851 1,1,
%e A167851 1,0,1,
%e A167851 1,0,0,1,
%e A167851 1,0,1,0,1,
%e A167851 1,0,1,1,0,1,
%e A167851 1,0,1,0,1,0,1,
%e A167851 1,0,1,1,1,1,0,1,
%e A167851 1,0,0,0,0,0,0,0,1,
%e A167851 1,1,1,1,1,1,1,1,1,1,
%e A167851 And so on...
%Y A167851 Cf. A159800, A164002, A167850, A167852.
%K A167851 easy,nonn,tabf,new
%O A167851 1,1
%A A167851 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167850
%S A167850 1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,
%T A167850 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,
%U A167850 1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1
%N A167850 Triangle read by rows in which each row contains a finite triangle as shown below.
%C A167850 See also A159800, A164002, A167851 and A167852.
%e A167850 The main triangle begins:
%e A167850 1,
%e A167850 1,1,1,
%e A167850 1,1,1,1,1,1,
%e A167850 1,1,1,1,0,1,1,1,1,1,
%e A167850 1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,
%e A167850 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1,1,1,1,
%e A167850 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,1,1,1,1,1,1,
%e A167850 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,
%e A167850 In which row 1 contains a vertex, only:
%e A167850 1,
%e A167850 Row 2 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 Row 3 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,1,1,
%e A167850 Row 4 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,1,1,1,
%e A167850 Row 5 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,1,1,1,1,
%e A167850 Row 6 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,0,0,0,1,
%e A167850 1,1,1,1,1,1,
%e A167850 Row 7 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,0,0,0,1,
%e A167850 1,0,0,0,0,1,
%e A167850 1,1,1,1,1,1,1,
%e A167850 Row 8 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,0,0,0,1,
%e A167850 1,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,1,
%e A167850 1,1,1,1,1,1,1,1,
%e A167850 Row 9 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,0,0,0,1,
%e A167850 1,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,0,1,
%e A167850 1,1,1,1,1,1,1,1,1,
%e A167850 Row 10 contains the triangle:
%e A167850 1,
%e A167850 1,1,
%e A167850 1,0,1,
%e A167850 1,0,0,1,
%e A167850 1,0,0,0,1,
%e A167850 1,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,0,1,
%e A167850 1,0,0,0,0,0,0,0,1,
%e A167850 1,1,1,1,1,1,1,1,1,1,
%e A167850 And so on...
%Y A167850 Cf. A159800, A164002, A167851, A167852.
%K A167850 easy,nonn,tabf,new
%O A167850 1,1
%A A167850 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167858
%S A167858 3,14,36,36,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A167858 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A167858 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167858 A000004 preceded by 3, 14, 36, 36, 12.
%C A167858 Inverse binomial transform of A166941/2.
%F A167858 a(0) = 3, a(1) = 14, a(2) = 36, a(3) = 36, a(4) = 12, a(n) = 0 for n > 4.
%F A167858 G.f.: 3+14*x+36*x^2+36*x^3+12*x^4.
%o A167858 (PARI) {concat([3, 14, 36, 36, 12], vector(98))}
%Y A167858 Cf. A000004 (zero sequence), A166941 (product plus sum of four consecutive nonnegative numbers), A166926 (1, 2, 4, 0, 0, 0, 0, ...), A130706 (1, 2, 0, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, 0, ...).
%K A167858 easy,nonn,new
%O A167858 0,1
%A A167858 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 13 2009

%I A167852
%S A167852 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,2,2,1,1,1,1,1,1,
%T A167852 1,1,1,1,2,1,1,2,2,1,1,2,2,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,1,1,2,3,2,
%U A167852 1,1,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,3,3,2,1
%N A167852 Triangle read by rows in which each row contains a finite triangle as shown below.
%C A167852 See also A159800, A164002, A167850 and A167851.
%e A167852 The main triangle begins:
%e A167852 1,
%e A167852 1,1,1,
%e A167852 1,1,1,1,1,1,
%e A167852 1,1,1,1,2,1,1,1,1,1,
%e A167852 1,1,1,1,2,1,1,2,2,1,1,1,1,1,1,
%e A167852 1,1,1,1,2,1,1,2,2,1,1,2,2,2,1,1,1,1,1,1,1,
%e A167852 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,2,2,2,1,1,1,1,1,1,1,1,
%e A167852 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,3,3,2,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1,
%e A167852 In which row 1 contains a vertex, only:
%e A167852 1,
%e A167852 Row 2 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 Row 3 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,1,1,
%e A167852 Row 4 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,1,1,1,
%e A167852 Row 5 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,1,1,1,1,
%e A167852 Row 6 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,2,2,2,1,
%e A167852 1,1,1,1,1,1,
%e A167852 Row 7 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,2,3,2,1,
%e A167852 1,2,2,2,2,1,
%e A167852 1,1,1,1,1,1,1,
%e A167852 Row 8 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,2,3,2,1,
%e A167852 1,2,3,3,2,1,
%e A167852 1,2,2,2,2,2,1,
%e A167852 1,1,1,1,1,1,1,1,
%e A167852 Row 9 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,2,3,2,1,
%e A167852 1,2,3,3,2,1,
%e A167852 1,2,3,3,3,2,1,
%e A167852 1,2,2,2,2,2,2,1,
%e A167852 1,1,1,1,1,1,1,1,1,
%e A167852 Row 10 contains the triangle:
%e A167852 1,
%e A167852 1,1,
%e A167852 1,2,1,
%e A167852 1,2,2,1,
%e A167852 1,2,3,2,1,
%e A167852 1,2,3,3,2,1,
%e A167852 1,2,3,4,3,2,1,
%e A167852 1,2,3,3,3,3,2,1,
%e A167852 1,2,2,2,2,2,2,2,1,
%e A167852 1,1,1,1,1,1,1,1,1,1,
%e A167852 And so on...
%Y A167852 Cf. A159800, A164002, A167850, A167851.
%K A167852 easy,nonn,tabf,new
%O A167852 1,15
%A A167852 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167857
%S A167857 1,2,3,5,7,9,10,11,13,17,19,22,23,25,29,31,34,37,41,43,46,47,49,53,55,
%T A167857 58,59,61,67,71,73,79,82,83,85,89,91,94,97,101,103,106,107,109,113,115,
%U A167857 118,121,127,131,133,137,139,142,145,149,151,157,163,166,167,169,171
%N A167857 Numbers whose divisors are represented by an integer polynomial.
%C A167857 That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.
%C A167857 Every prime has this property, as do 1 and 9, the squares of primes of the form 6k+1, and semiprimes p*q with p and q both primes of the form 3k-1 or 3k+1. Terms of the form p^2*q also appear. We can find terms of the form p^m for any m. For example, 2311^13 is the smallest 13th power that appears. For any m, it seems that p^m appears for p a prime of the form k*m#+1, where m# is the product of the primes up to m. Are there terms with three distinct prime divisors?
%e A167857 The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.
%t A167857 Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]
%Y A167857 Cf. A108164, A108166, A112774 (forms of semiprimes)
%Y A167857 Cf. 002476 (primes of the form 6k+1)
%Y A167857 Cf. A132230 (primes of the form 30k+1)
%Y A167857 Cf. A073103 (primes of the form 210k+1)
%Y A167857 Cf. A073917 (least prime of the form k*prime(n)#+1)
%K A167857 nonn,new
%O A167857 1,2
%A A167857 T. D. Noe (noe(AT)sspectra.com), Nov 13 2009

%I A167853
%S A167853 2,3,5,7,131,151,163,173,181,191,193
%N A167853 Generalized mountain primes.
%C A167853 Primes in A134853.
%C A167853 a(1) to a(4) are equal to A000040. For n>4 the structure of the digits represent a mountain. The first digits are in increasing order. The last digits are in decreasing order. The numbers have only a largest digit that represent the top of the mountain. This sequence is finite.
%C A167853 See also A134951, mountain primes.
%Y A167853 Cf. A000040, A134853, A134941, A134951, A135642, A167845.
%K A167853 base,fini,more,nonn,new
%O A167853 1,1
%A A167853 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009, Nov 15 2009

%I A167856
%S A167856 2,4,6,10,12,18,20,24,30,42,60
%N A167856 Values of n that produce record values of a(n) in A167401
%Y A167856 Cf. A167401
%K A167856 more,nonn,new
%O A167856 1,1
%A A167856 J. Lowell (jhbubby(AT)mindspring.com), Nov 13 2009

%I A167845
%S A167845 131,151,157,163,167,173,179,181
%N A167845 Concave primes.
%C A167845 Primes in A135642.
%C A167845 Primes whose structure of digits represent a concave function or a concave object. In the graphic representation the points are connected by imaginary line segments (or line curves) from left to right.
%Y A167845 Cf. A134811, A134951, A134971, A135642, A167841, A167842, A167843, A167844, A167846, A167853.
%K A167845 base,more,nonn,new
%O A167845 1,1
%A A167845 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167846
%S A167846 1021,1031,1033,1051,1061,1063,1069
%N A167846 Concave-convex primes.
%C A167846 Primes in A163278.
%C A167846 Primes numbers with more than three digits that are not straight-line numbers (A135643), concave numbers (A135642) or convex numbers (A135641).
%Y A167846 Cf. A134811, A134951, A134971, A135641, A135642, A135643, A163278, A167841, A167842, A167843, A167844, A167845, A167853.
%K A167846 base,more,nonn,new
%O A167846 1,1
%A A167846 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167844
%S A167844 101,103,107,109,113,127,137,139,149,211
%N A167844 Convex primes.
%C A167844 Primes in A135641.
%C A167844 Primes whose structure of digits represent a convex function or a convex object. In the graphic representation the points are connected by imaginary line segments (or line curves) from left to right.
%Y A167844 Cf. A134811, A134951, A134971, A135641, A167841, A167842, A167843, A167845, A167846, A167853.
%K A167844 base,more,nonn,new
%O A167844 1,1
%A A167844 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167843
%S A167843 113,127,137,139,149,157,167,179,199,211,223
%N A167843 Obtuse-angled primes.
%C A167843 Primes in A135603.
%C A167843 Primes whose structure of digits represent an obtuse angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right.
%Y A167843 Cf. A134811, A134951, A134971, A135603, A167841, A167842, A167844, A167845, A167846, A167853.
%K A167843 base,more,nonn,new
%O A167843 1,1
%A A167843 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167842
%S A167842 101,787
%N A167842 Right-angled primes.
%C A167842 Primes in A135602.
%C A167842 Primes whose structure of digits represent a right angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite.
%Y A167842 Cf. A134811, A134951, A134971, A135602, A167841, A167843, A167844, A167845, A167846, A167853.
%K A167842 base,bref,more,nonn,new
%O A167842 1,1
%A A167842 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167841
%S A167841 103,107,109,131,151,163,173,181,191,193
%N A167841 Acute-angled primes.
%C A167841 Primes in A135601.
%C A167841 Primes whose structure of digits represent an acute angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite.
%Y A167841 Cf. A134811, A134951, A134971, A135601, A167842, A167843, A167844, A167845, A167846, A167853.
%K A167841 base,more,nonn,new
%O A167841 1,1
%A A167841 Omar E. Pol (info(AT)polprimos.com), Nov 13 2009

%I A167840
%S A167840 2,1847,2179,2503,2819,3137,3433,3967,4177,4621,4831,5039,5623
%N A167840 Six-times-isolated primes: primes p such that neither p+-2, p+-4, p+-6, p+-8, p+-10 nor p+-12 is prime.
%e A167840 a(1)=2 (-10,-8,-6,-4,-2,0,4,6,8,10,12 are nonprimes); a(2)=1847 (1835,1837,1839,1841,1843,1845,1849,1851,1853,1855,1857,1859 are nonprimes).
%Y A167840 Cf. A000040, A167771.
%K A167840 nonn,more,new
%O A167840 1,1
%A A167840 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 13 2009

%I A167839
%S A167839 2,211,1511,1831,2179,2503,2579,2633,2819,2939,3137,3271,3433,3659,3967,
%T A167839 3967,3989,4111,4177,4409,4621,4691,4831,4889,5039,5623
%N A167839 Five-times-isolated primes: primes p such that neither p+-2, p+-4, p+-6, p+-8 nor p+-10 is prime.
%e A167839 a(1)=2 (-8,-6,-4,-2,0,4,6,8,10,12 are nonprimes); a(2)=211 (201,203,205,207,209,213,215,217,219,221 are nonprimes).
%Y A167839 Cf. A000040, A167771.
%K A167839 nonn,more,new
%O A167839 1,1
%A A167839 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 13 2009

%I A160672
%S A160672 1,1,1,1,1,5,5,35,35,5,175,1925,175,3575,175175,875875,17875,14889875,
%T A160672 14889875,282907625,202076875,9901766875,108919435625,4647768125,
%U A160672 2505147019375,62628675484375,814172781296875,874514265625
%N A160672 Denominator of Laguerre(n, 12).
%Y A160672 For numerators see A160671.
%K A160672 nonn,frac,new
%O A160672 0,6
%A A160672 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009 >

%I A160671
%S A160671 1,11,49,107,97,137,427,1249,2147,329,1601,129559,9329,31523,10326263,
%T A160671 51307231,255571,557933137,913977949,12806652259,565484693,388623944407,
%U A160671 6314613012857,212124740033,27554289038383,1724536942646717
%V A160671 1,-11,49,-107,97,137,-427,-1249,2147,329,-1601,-129559,-9329,31523,10326263,51307231,
%W A160671 255571,-557933137,-913977949,-12806652259,-565484693,388623944407,6314613012857,
%X A160671 212124740033,27554289038383,-1724536942646717,-42239185706105669,-45706055907059
%N A160671 Numerator of Laguerre(n, 12).
%Y A160671 For denominators see A160672.
%K A160671 sign,frac,new
%O A160671 0,2
%A A160671 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160667
%S A160667 1,1,2,3,24,60,144,2520,40320,36288,518400,362880,43545600,283046400,
%T A160667 1585059840,59439744000,380414361600,2309658624000,582033973248000,
%U A160667 1105864549171200,221172909834240000,464463110651904000
%N A160667 Denominator of Laguerre(n, 11).
%Y A160667 For numerators see A160655.
%K A160667 nonn,frac,new
%O A160667 0,3
%A A160667 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160655
%S A160655 1,10,79,217,1025,2447,5125,113717,345889,1663111,15004553,5570839,
%T A160655 1823851261,8755954747,5608062241,1968529931257,14486425445831,
%U A160655 41829962445559,6878017387517971,36856916172852287,7836972251504652829
%V A160655 1,-10,79,-217,1025,2447,-5125,-113717,345889,1663111,15004553,-5570839,-1823851261,
%W A160655 -8755954747,5608062241,1968529931257,14486425445831,41829962445559,-6878017387517971,
%X A160655 -36856916172852287,-7836972251504652829,-8768150285373043687,59117879235834356159
%N A160655 Numerator of Laguerre(n, 11).
%Y A160655 For denominators see A160667.
%K A160655 sign,frac,new
%O A160655 0,2
%A A160655 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160654
%S A160654 1,1,1,3,1,3,9,21,63,567,63,6237,18711,81081,1702701,464373,1702701,
%T A160654 86837751,71049069,549972423,14849255421,23987258757,1143392667417,
%U A160654 78894094051773,236682282155319,78894094051773,279715424365377
%N A160654 Denominator of Laguerre(n, 10).
%Y A160654 For numerators see A160653.
%K A160654 nonn,frac,new
%O A160654 0,4
%A A160654 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160653
%S A160653 1,9,31,137,11,103,31,649,1027,8387,1763,90667,185309,2014549,35712389,
%T A160653 1568327,25932791,2065388131,1325044249,2183157953,177616901779,
%U A160653 514233019213,23712818592973,872068873818067,670711846972459
%V A160653 1,-9,31,-137,11,103,-31,-649,-1027,8387,1763,90667,-185309,-2014549,-35712389,
%W A160653 -1568327,25932791,2065388131,1325044249,2183157953,-177616901779,-514233019213,
%X A160653 -23712818592973,-872068873818067,670711846972459,1185956279291023,5868401189931107
%N A160653 Numerator of Laguerre(n, 10).
%Y A160653 For denominators see A160654.
%K A160653 sign,frac,new
%O A160653 0,2
%A A160653 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160641
%S A160641 1,1,2,1,8,5,80,140,4480,560,44800,123200,1971200,1601600,358758400,
%T A160641 448448000,2207744000,60988928000,75063296000,4635158528000,
%U A160641 370812682240000,81115274240000,57105153064960000,25258048471040000
%N A160641 Denominator of Laguerre(n, 9).
%Y A160641 For numerators see A160640.
%K A160641 nonn,frac,new
%O A160641 0,3
%A A160641 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160640
%S A160640 1,8,47,26,37,104,863,1633,81479,3247,473551,2070053,19541099,5299537,
%T A160640 4828712387,6662908309,17244807859,180304757051,874291731563,
%U A160640 66578605064561,3886381511522161,185817086098037,369129092519093411
%V A160640 1,-8,47,-26,-37,104,863,-1633,-81479,-3247,473551,2070053,19541099,-5299537,
%W A160640 -4828712387,-6662908309,-17244807859,180304757051,874291731563,66578605064561,
%X A160640 3886381511522161,185817086098037,-369129092519093411,-310895814618726151
%N A160640 Numerator of Laguerre(n, 9).
%Y A160640 For denominators see A160641.
%K A160640 sign,frac,new
%O A160640 0,2
%A A160640 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160639
%S A160639 1,1,1,3,3,15,45,315,45,2835,14175,155925,66825,6081075,42567525,
%T A160639 91216125,638512875,10854718875,97692469875,2228279625,9280784638125,
%U A160639 14992036723125,18015640768125,49308808782358125,147926426347074375
%N A160639 Denominator of Laguerre(n, 8).
%Y A160639 For numerators see A160635.
%K A160639 nonn,frac,new
%O A160639 0,4
%A A160639 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160635
%S A160635 1,7,17,37,29,119,541,563,403,29893,50161,842557,668137,49176251,
%T A160635 71971349,472553467,5767166357,91253170621,398562264797,3871198033,
%U A160635 60961731099469,129943563856429,135463017394157,187641213694147637
%V A160635 1,-7,17,-37,-29,119,541,563,-403,-29893,-50161,842557,668137,49176251,71971349,
%W A160635 -472553467,-5767166357,-91253170621,-398562264797,3871198033,60961731099469,
%X A160635 129943563856429,135463017394157,187641213694147637,-151190634238477433
%N A160635 Numerator of Laguerre(n, 8).
%Y A160635 For denominators see A160639.
%K A160635 sign,frac,new
%O A160635 0,2
%A A160635 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160634
%S A160634 1,1,2,3,8,60,720,120,5760,25920,172800,2851200,68428800,49420800,
%T A160634 161740800,13343616000,142331904000,3629463552000,130660687872000,
%U A160634 31827603456000,49651061391360000,74476592087040000,40456420392960000
%N A160634 Denominator of Laguerre(n, 7).
%Y A160634 For numerators see A160633.
%K A160634 nonn,frac,new
%O A160634 0,3
%A A160634 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160633
%S A160633 1,6,23,11,69,31,4927,757,1847,136109,1138741,10302661,83677831,
%T A160633 248520407,978257053,55741960849,84809304859,10962379087843,
%U A160633 687422896231961,173321349205597,184450422077432191,61692018244147721
%V A160633 1,-6,23,-11,-69,-31,4927,757,1847,-136109,-1138741,-10302661,83677831,248520407,
%W A160633 978257053,55741960849,84809304859,-10962379087843,-687422896231961,-173321349205597,
%X A160633 -184450422077432191,-61692018244147721,88624089843204841,166251411098859747263
%N A160633 Numerator of Laguerre(n, 7).
%Y A160633 For denominators see A160634.
%K A160633 sign,frac,new
%O A160633 0,2
%A A160633 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160632
%S A160632 1,1,1,1,1,5,1,35,35,1,175,385,1925,25025,35035,875875,5005,14889875,
%T A160632 14889875,11316305,1414538125,1980353375,108919435625,357878145625,
%U A160632 26369968625,62628675484375,162834556259375,814172781296875
%N A160632 Denominator of Laguerre(n, 6).
%Y A160632 For numerators see A160631.
%K A160632 nonn,frac,new
%O A160632 0,6
%A A160632 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160631
%S A160631 1,5,7,1,5,19,1,149,137,1,389,1517,6823,38617,34213,2572987,18391,
%T A160631 45729683,22001999,5642893,3149527091,6408414379,361288088869,
%U A160631 905150581043,30112639363,29055165817369,309542042395973
%V A160631 1,-5,7,1,-5,-19,1,149,137,1,-389,-1517,-6823,-38617,34213,2572987,18391,45729683,
%W A160631 22001999,-5642893,-3149527091,-6408414379,-361288088869,-905150581043,-30112639363,
%X A160631 29055165817369,309542042395973,2330434589510479,18100779790639279,93274917955709869
%N A160631 Numerator of Laguerre(n, 6).
%Y A160631 For denominators see A160632.
%K A160631 sign,frac,new
%O A160631 0,2
%A A160631 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160630
%S A160630 1,1,2,3,24,6,144,126,8064,2592,145152,24948,19160064,8895744,
%T A160630 3487131648,1307674368,23911759872,711374856192,51218989645824,
%U A160630 5529322745856,556091887583232,20436376868683776,1798401164444172288
%N A160630 Denominator of Laguerre(n, 5).
%Y A160630 For numerators see A160629.
%K A160630 nonn,frac,new
%O A160630 0,3
%A A160630 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160629
%S A160629 1,4,7,8,31,19,301,41,18029,6977,254927,2683,27755351,20708341,
%T A160629 8065683737,1998225857,7525030013,654365168147,93746955893939,
%U A160629 12226393924361,1123431147590407,27739255684444921,748334911424041583
%V A160629 1,-4,7,8,-31,-19,-301,41,18029,6977,254927,2683,-27755351,-20708341,-8065683737,
%W A160629 -1998225857,-7525030013,654365168147,93746955893939,12226393924361,1123431147590407,
%X A160629 27739255684444921,748334911424041583,-424460759787923281,-73378993429711239701
%N A160629 Numerator of Laguerre(n, 5).
%Y A160629 For denominators see A160630.
%K A160629 sign,frac,new
%O A160629 0,2
%A A160629 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160628
%S A160628 1,1,1,3,1,15,45,35,315,2835,675,155925,467775,2027025,42567525,
%T A160628 638512875,70945875,1550674125,7514805375,618718975875,9280784638125,
%U A160628 194896477400625,714620417135625,49308808782358125,21132346621010625
%N A160628 Denominator of Laguerre(n, 4).
%Y A160628 For numerators see A160627.
%K A160628 nonn,frac,new
%O A160628 0,4
%A A160628 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160627
%S A160627 1,3,1,7,1,13,83,57,197,1543,931,255217,620863,1283437,8184347,
%T A160627 581939153,96325549,2261420263,9237901697,466203083257,1399462769491,
%U A160627 88080802505933,675385729846153,61756890564641033,28280520375244993
%V A160627 1,-3,1,7,1,-13,-83,-57,-197,1543,931,255217,620863,1283437,-8184347,-581939153,
%W A160627 -96325549,-2261420263,-9237901697,-466203083257,-1399462769491,88080802505933,
%X A160627 675385729846153,61756890564641033,28280520375244993,495763088616446491
%N A160627 Numerator of Laguerre(n, 4).
%Y A160627 For denominators see A160628.
%K A160627 sign,frac,new
%O A160627 0,2
%A A160627 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160626
%S A160626 1,1,2,1,8,20,80,280,4480,2240,44800,246400,1971200,12812800,358758400,
%T A160626 81536000,28700672000,243955712000,88711168000,9270317056000,
%U A160626 370812682240000,117985853440000,3359126650880000,656709260247040000
%N A160626 Denominator of Laguerre(n, 3).
%Y A160626 For numerators see A160625.
%K A160626 nonn,frac,new
%O A160626 0,3
%A A160626 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160625
%S A160625 1,2,1,1,11,17,1,209,4967,2377,31361,44549,670907,9518329,343496509,
%T A160625 78783701,22768521641,119672922599,10898629213,2270181550733,
%U A160625 206731528257599,91509891412157,2949362928754771,565723903687129379
%V A160625 1,-2,-1,1,11,17,-1,-209,-4967,-2377,-31361,-44549,670907,9518329,343496509,78783701,
%W A160625 22768521641,119672922599,10898629213,-2270181550733,-206731528257599,-91509891412157,
%X A160625 -2949362928754771,-565723903687129379,-7753360902058576583,-69708311239418420641
%N A160625 Numerator of Laguerre(n, 3).
%Y A160625 For denominators see A160626.
%K A160625 sign,frac,new
%O A160625 0,2
%A A160625 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160624
%S A160624 1,1,1,3,3,15,45,315,315,2835,14175,155925,467775,6081075,42567525,
%T A160624 638512875,58046625,10854718875,8881133625,1856156927625,9280784638125,
%U A160624 194896477400625,2143861251406875,3792985290950625,147926426347074375
%N A160624 Denominator of Laguerre(n, 2).
%Y A160624 For numerators see A160623.
%K A160624 nonn,frac,new
%O A160624 0,4
%A A160624 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160623
%S A160623 1,1,1,1,1,11,37,209,113,23,4381,84389,310517,4103887,25059901,
%T A160623 274436401,13182829,104362273,1748375381,690031209911,4647089032189,
%U A160623 112233351264271,1276100569319881,2131681036523177,71497025649480187
%V A160623 1,-1,-1,-1,1,11,37,209,113,23,-4381,-84389,-310517,-4103887,-25059901,-274436401,
%W A160623 -13182829,-104362273,1748375381,690031209911,4647089032189,112233351264271,
%X A160623 1276100569319881,2131681036523177,71497025649480187,1365106755339875117
%N A160623 Numerator of Laguerre(n, 2).
%Y A160623 For denominators see A160624.
%K A160623 sign,frac,new
%O A160623 0,6
%A A160623 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160622
%S A160622 1,1,2,3,8,15,144,420,5760,4536,403200,1995840,6220800,259459200,
%T A160622 1341204480,46702656000,1394852659200,11115232128000,376610217984000,
%U A160622 96543730483200,128047474114560000,1277273554292736000
%N A160622 Denominator of Laguerre(n, 1).
%Y A160622 For numerators see A160621.
%K A160622 nonn,frac,new
%O A160622 0,3
%A A160622 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160621
%S A160621 1,0,1,2,5,7,37,17,887,1405,168919,958271,3086837,122693929,559876951,
%T A160621 15779421743,337767590383,1531923385313,11912361112367,6819537030283,
%U A160621 21032925955607701,313699426574980519,16703816669710968821
%V A160621 1,0,-1,-2,-5,-7,-37,-17,887,1405,168919,958271,3086837,122693929,559876951,
%W A160621 15779421743,337767590383,1531923385313,11912361112367,-6819537030283,
%X A160621 -21032925955607701,-313699426574980519,-16703816669710968821,-212752402370938916881
%N A160621 Numerator of Laguerre(n, 1).
%Y A160621 For denominators see A160622.
%K A160621 sign,frac,new
%O A160621 0,4
%A A160621 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160618
%S A160618 1,1,2,3,24,60,720,2520,40320,25920,3628800,19958400,479001600,
%T A160618 3113510400,87178291200,59439744000,426995712000,177843714048000,
%U A160618 582033973248000,60822550204416000,143111882833920000
%N A160618 Denominator of Laguerre(n, -1).
%Y A160618 For numerators see A160617.
%K A160618 nonn,frac,new
%O A160618 0,3
%A A160618 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160617
%S A160617 1,2,7,17,209,773,13327,65461,1441729,1255151,234662231,1702678841,
%T A160617 53334454417,448162154317,16083557845279,13946689584823,126523856174033,
%U A160617 66120494322107921,269906478537389909,34987413853951524577
%N A160617 Numerator of Laguerre(n, -1).
%Y A160617 For denominators see A160618.
%K A160617 nonn,frac,new
%O A160617 0,2
%A A160617 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160616
%S A160616 1,1,1,3,1,15,45,105,315,405,4725,155925,467775,96525,42567525,
%T A160616 638512875,30405375,10854718875,97692469875,618718975875,189403768125,
%U A160616 194896477400625,238206805711875,7044115540336875,8701554491004375
%N A160616 Denominator of Laguerre(n, -2).
%Y A160616 For numerators see A160615.
%K A160616 nonn,frac,new
%O A160616 0,4
%A A160616 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160615
%S A160615 1,3,7,43,27,719,3661,13991,66769,133261,2363513,116441047,513267739,
%T A160615 153434147,96790969339,2053217625931,136839921293,67725860135459,
%U A160615 837687671342383,7232743280136193,2996031500521181,4142815387557270467
%N A160615 Numerator of Laguerre(n, -2).
%Y A160615 For denominators see A160616.
%K A160615 nonn,frac,new
%O A160615 0,2
%A A160615 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160614
%S A160614 1,1,2,1,8,10,80,35,4480,1120,6400,61600,1971200,6406400,358758400,
%T A160614 112112000,28700672000,17425408000,975822848000,2317579264000,
%U A160614 370812682240000,49917091840000,57105153064960000,41044328765440000
%N A160614 Denominator of Laguerre(n, -3).
%Y A160614 For numerators see A160613.
%K A160614 nonn,frac,new
%O A160614 0,3
%A A160614 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160613
%S A160613 1,4,23,28,491,1249,19223,15476,3515161,1512661,14496817,228800107,
%T A160613 11770539419,60428965661,5262254717509,2521163372543,976843770850217,
%U A160613 887131806309703,73511154681979031,255777165814872577
%N A160613 Numerator of Laguerre(n, -3).
%Y A160613 For denominators see A160614.
%K A160613 nonn,frac,new
%O A160613 0,2
%A A160613 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160612
%S A160612 1,1,1,3,3,15,9,315,315,567,14175,6237,467775,6081075,773955,638512875,
%T A160612 9823275,10854718875,7514805375,21837140325,9280784638125,
%U A160612 38979295480125,2143861251406875,3792985290950625,1183411410776595
%N A160612 Denominator of Laguerre(n, -4).
%Y A160612 For numerators see A160611.
%K A160612 nonn,frac,new
%O A160612 0,4
%A A160612 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160611
%S A160611 1,5,17,143,355,4043,5177,367271,713723,2410003,109669391,85569361,
%T A160611 11122330591,245535162239,52108328723,70514170732823,1753034045867,
%U A160611 3087820148584967,3365163124738543,15216530369586809
%N A160611 Numerator of Laguerre(n, -4).
%Y A160611 For denominators see A160612.
%K A160611 nonn,frac,new
%O A160611 0,2
%A A160611 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160610
%S A160610 1,1,2,3,8,12,144,56,8064,36288,48384,798336,19160064,41513472,
%T A160610 3487131648,5230697472,6199345152,1422749712384,51218989645824,
%U A160610 162193467211776,3892643213082624,3144057979797504,599467054814724096
%N A160610 Denominator of Laguerre(n, -5).
%Y A160610 For numerators see A160609.
%K A160610 nonn,frac,new
%O A160610 0,3
%A A160610 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160609
%S A160609 1,6,47,223,1643,6187,173339,148591,44999149,409402223,1067091709,
%T A160609 33428532191,1484643512329,5818648016477,866275665579983,
%U A160609 2261064256234247,4585711907324687,1773637758861597199
%N A160609 Numerator of Laguerre(n, -5).
%Y A160609 For denominators see A160610.
%K A160609 nonn,frac,new
%O A160609 0,2
%A A160609 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160608
%S A160608 1,1,1,1,1,5,5,35,5,35,175,1925,275,25025,175175,125125,875875,14889875,
%T A160608 14889875,282625,1414538125,9901766875,2222845625,2505147019375,
%U A160608 2505147019375,62628675484375,116310397328125,814172781296875
%N A160608 Denominator of Laguerre(n, -6).
%Y A160608 For numerators see A160607.
%K A160608 nonn,frac,new
%O A160608 0,6
%A A160608 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160607
%S A160607 1,7,31,109,331,4529,11453,190433,61391,928943,9677971,214858067,
%T A160607 60236303,10492980947,137504412401,180206776249,2272545257401,
%U A160607 68446106098751,119418042814439,3899046884359,33130663362484669
%N A160607 Numerator of Laguerre(n, -6).
%Y A160607 For denominators see A160608.
%K A160607 nonn,frac,new
%O A160607 0,2
%A A160607 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160606
%S A160606 1,1,2,3,24,15,720,180,5760,6480,518400,1425600,68428800,27799200,
%T A160606 1779148800,6671808000,426995712000,907365888000,130660687872000,
%U A160606 620638267392000,49651061391360000,9309574010880000,3276970051829760000
%N A160606 Denominator of Laguerre(n, -7).
%Y A160606 For numerators see A160605.
%K A160606 nonn,frac,new
%O A160606 0,3
%A A160606 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160605
%S A160605 1,8,79,458,12113,22394,2921911,1856741,142939831,369403781,65257860137,
%T A160605 382924264759,38054710694503,31162073928841,3924727189601143,
%U A160605 28342240146349903,3424956884584928081,13496977684191852821
%N A160605 Numerator of Laguerre(n, -7).
%Y A160605 For denominators see A160606.
%K A160605 nonn,frac,new
%O A160605 0,2
%A A160605 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160604
%S A160604 1,1,1,3,1,15,45,105,315,405,525,155925,467775,2027025,42567525,
%T A160604 58046625,30405375,10854718875,8881133625,206239658625,9280784638125,
%U A160604 17717861581875,714620417135625,7044115540336875,147926426347074375
%N A160604 Denominator of Laguerre(n, -8).
%Y A160604 For numerators see A160603.
%K A160604 nonn,frac,new
%O A160604 0,4
%A A160604 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160603
%S A160603 1,9,49,619,737,35111,305917,1930753,14779003,46162429,139399997,
%T A160603 92993347501,606843495583,5559358281401,240588034396789,660664176177209,
%U A160603 682418523335551,471285453584720627,732894385728160361
%N A160603 Numerator of Laguerre(n, -8).
%Y A160603 For denominators see A160604.
%K A160603 nonn,frac,new
%O A160603 0,2
%A A160603 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160602
%S A160602 1,1,2,1,8,20,16,280,4480,64,44800,49280,1971200,1830400,14350336,
%T A160602 896896000,74547200,243955712000,6823936000,109062553600,52973240320000,
%U A160602 51913775513600,3359126650880000,93815608606720000,2101469632790528000
%N A160602 Denominator of Laguerre(n, -9).
%Y A160602 For numerators see A160601.
%K A160602 nonn,frac,new
%O A160602 0,3
%A A160602 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160601
%S A160601 1,10,119,271,8315,70499,174139,8731657,376455481,13762207,23585826391,
%T A160601 61123934273,5568591324683,11428044042013,192745690532125,
%U A160601 25291642192372513,4316709625044329,28425936917236036637
%N A160601 Numerator of Laguerre(n, -9).
%Y A160601 For denominators see A160602.
%K A160601 nonn,frac,new
%O A160601 0,2
%A A160601 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160589
%S A160589 1,1,1,3,3,3,9,63,63,567,81,6237,1701,243243,1702701,5108103,5108103,
%T A160589 1772199,781539759,14849255421,14849255421,311834363841,3430178002251,
%U A160589 7172190368343,33811754593617,236682282155319,3076869668019147
%N A160589 Denominator of Laguerre(n, -10).
%Y A160589 For numerators see A160587.
%K A160589 nonn,frac,new
%O A160589 0,4
%A A160589 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160587
%S A160587 1,11,71,1043,4273,15403,151049,3196873,9065053,219181447,80314181,
%T A160587 15236362807,9881224121,3255762225533,51047003351891,334353686524303,
%U A160587 713211624791333,516702875135479,466605840054517979
%N A160587 Numerator of Laguerre(n, -10).
%Y A160587 For denominators see A160589.
%K A160587 nonn,frac,new
%O A160587 0,2
%A A160587 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160586
%S A160586 1,1,2,3,8,30,720,210,40320,90720,1209600,226800,43545600,5241600,
%T A160586 7925299200,14859936000,634023936000,8083805184000,582033973248000,
%U A160586 115194223872000,221172909834240000,1161157776629760000
%N A160586 Denominator of Laguerre(n, -11).
%Y A160586 For numerators see A160566.
%K A160586 nonn,frac,new
%O A160586 0,3
%A A160586 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160566
%S A160566 1,12,167,1312,15243,218149,18053887,16744993,9564235169,60519568871,
%T A160566 2162548405277,1042027382717,495498798823907,142936348776673,
%U A160566 502871478378225709,2136147813028051787,201533194180774871969
%N A160566 Numerator of Laguerre(n, -11).
%Y A160566 For denominators see A160586.
%K A160566 nonn,frac,new
%O A160566 0,2
%A A160566 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160555
%S A160555 1,1,1,1,1,5,5,35,35,35,175,1925,1925,25025,13475,79625,875875,14889875,
%T A160555 1353625,282907625,1414538125,9901766875,108919435625,2505147019375,
%U A160555 2505147019375,4817590421875,6728700671875,62628675484375
%N A160555 Denominator of Laguerre(n, -12).
%Y A160555 For numerators see A160554.
%K A160555 nonn,frac,new
%O A160555 0,6
%A A160555 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160554
%S A160554 1,13,97,541,2497,50273,182309,4256087,13247651,38903687,543392719,
%T A160554 15986775377,41148884933,1330667437997,1728533404931,23966754421349,
%U A160554 603184673218147,22925161516659191,4561425357186553,2045955360223686629
%N A160554 Numerator of Laguerre(n, -12).
%Y A160554 For denominators see A160555.
%K A160554 nonn,frac,new
%O A160554 0,2
%A A160554 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160553
%S A160553 36,37,55,70,79,84,93,99,105,111,118,128,134,138,140,149,156,161,163,168,
%T A160553 174,180,185,199
%N A160553 Numbers n such that p(49n+47) == 0 mod 343 but n is not of the form 7k+4, where p() = A000041().
%C A160553 Watson found the terms 36,37,55 via A002300. I extended the sequence to 199 using his method (but with Maple's help). - N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009
%D A160553 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See p. 128.
%H A160553 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%K A160553 nonn,more,new
%O A160553 1,1
%A A160553 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160549
%S A160549 0,1,5,20,70,221,646,1772,4614,11490,27537,63808,143514,314279,671872,1405260,
%T A160549 2881030,5799093,11476452,22357584,42922558,81284699,151974124,280739800,
%U A160549 512761178,926568075,1657448779,2936506316,5155349836,8972488674,15487146900
%N A160549 Omit first term from A160534 and divide by 7.
%C A160549 These are Watson's coefficients beta'_n on page 125.
%D A160549 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160549 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%K A160549 nonn,new
%O A160549 0,3
%A A160549 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160539
%S A160539 1,7,35,140,490,1547,4522,12404,32298,80430,192759,446656,1004598,2199953,
%T A160539 4703104,9836820,20167210,40593651,80335164,156503088,300457906,568992893,
%U A160539 1063818868,1965178600,3589328246,6485976525,11602141453,20555544212,36087448852
%N A160539 Coefficients in the expansion of C/B^7, in Watson's notation of page 118.
%D A160539 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160539 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160539 See Maple code in A160525 for formula.
%e A160539 1+7*x^24+35*x^48+140*x^72+490*x^96+1547*x^120+4522*x^144+...
%K A160539 nonn,new
%O A160539 0,2
%A A160539 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160535
%S A160535 0,1,2,1,7,3,5,6,8,17,15,10,21,21,19,24,33,36,22,45,63,1,92,82,85,97,105,
%T A160535 82,28,58,120,120,210,122,180,3,231,138,225,168,255,210,5,282,294,219,284,
%U A160535 276,341,43,310,288,441,346,410,366,29,360,668,435,504,465,600,46,603,504
%V A160535 0,-1,2,1,-7,3,5,6,-8,-17,15,-10,21,21,-19,-24,-33,36,-22,45,63,1,-92,-82,85,-97,105,
%W A160535 82,28,-58,-120,120,-210,122,180,3,-231,-138,225,-168,255,210,5,-282,-294,219,-284,
%X A160535 276,341,-43,-310,-288,441,-346,410,366,-29,-360,-668,435,-504,465,600,46,-603,-504
%N A160535 Omit first term from A160534 and divide by 7.
%C A160535 These are Watson's coefficients beta_n on page 125.
%D A160535 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160535 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%K A160535 sign,new
%O A160535 0,3
%A A160535 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160534
%S A160534 1,7,14,7,49,21,35,42,56,119,105,70,147,147,133,168,231,252,154,315,441,
%T A160534 7,644,574,595,679,735,574,196,406,840,840,1470,854,1260,21,1617,966,1575,
%U A160534 1176,1785,1470,35,1974,2058,1533,1988,1932,2387,301,2170,2016,3087,2422
%V A160534 1,-7,14,7,-49,21,35,42,-56,-119,105,-70,147,147,-133,-168,-231,252,-154,315,441,
%W A160534 7,-644,-574,595,-679,735,574,196,-406,-840,840,-1470,854,1260,21,-1617,-966,1575,
%X A160534 -1176,1785,1470,35,-1974,-2058,1533,-1988,1932,2387,-301,-2170,-2016,3087,-2422
%N A160534 Coefficients in the expansion of B^7/C, in Watson's notation of page 118.
%D A160534 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160534 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160534 See Maple code in A160525 for formula.
%e A160534 1-7*x^24+14*x^48+7*x^72-49*x^96+21*x^120+35*x^144+42*x^168-...
%K A160534 sign,new
%O A160534 0,2
%A A160534 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160533
%S A160533 1,6,27,98,315,918,2492,6367,15495,36145,81326,177219,375461,775544,1565870,
%T A160533 3096615,6008917,11458720,21502964,39754385,72485518,130464603,231989748,
%U A160533 407847488,709365160,1221364655,2082872680,3519963776,5897536697,9800358525
%N A160533 Coefficients in the expansion of C^5/B^6, in Watson's notation of page 118.
%D A160533 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160533 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160533 See Maple code in A160525 for formula.
%e A160533 x^29+6*x^53+27*x^77+98*x^101+315*x^125+918*x^149+2492*x^173+...
%K A160533 nonn,new
%O A160533 0,2
%A A160533 N. J. A. Sloane (njas(AT)research.att.com), Nov 14 2009

%I A160528
%S A160528 1,5,20,65,190,506,1265,2986,6745,14645,30767,62745,124706,242110,460337,
%T A160528 858673,1574140,2839862,5048435,8852562,15327290,26224173,44372688,74301095,
%U A160528 123200079,202394897,329596348,532299955,852914900,1356426196,2141819621
%N A160528 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 118.
%D A160528 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160528 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160528 See Maple code in A160525 for formula.
%e A160528 x^23+5*x^47+20*x^71+65*x^95+190*x^119+506*x^143+1265*x^167+...
%Y A160528 Cf. A002300, A160525.
%K A160528 nonn,new
%O A160528 0,2
%A A160528 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160527
%S A160527 1,4,14,40,105,252,574,1237,2568,5138,9988,18893,34937,63238,112370,196244,
%T A160527 337477,572024,956956,1581321,2583637,4176495,6684820,10599939,16661401,
%U A160527 25972485,40171474,61672695,94017765,142368024,214211760,320350725,476299978
%N A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.
%D A160527 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160527 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160527 See Maple code in A160525 for formula.
%e A160527 x^17+4*x^41+14*x^65+40*x^89+105*x^113+252*x^137+574*x^161+...
%K A160527 nonn,new
%O A160527 0,2
%A A160527 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160526
%S A160526 1,3,9,22,51,108,221,427,804,1461,2596,4497,7652,12767,20984,33958,54255,
%T A160526 85580,133520,206066,315010,477083,716494,1067316,1578102,2316569,3377965,
%U A160526 4894045,7047970,10091120,14369439,20354090,28687663,40239129,56183879
%N A160526 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 118.
%D A160526 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160526 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160526 See Maple code in A160525 for formula.
%e A160526 x^11+3*x^35+9*x^59+22*x^83+51*x^107+108*x^131+221*x^155+...
%K A160526 nonn,new
%O A160526 0,2
%A A160526 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160525
%S A160525 1,2,5,10,20,36,65,109,183,295,471,732,1129,1705,2554,3769,5517,7979,11458,
%T A160525 16289,23007,32227,44869,62028,85284,116530,158432,214228,288348,386224,
%U A160525 515156,684109,904963,1192353,1565383,2047642,2669591,3468797,4493351,5802533
%N A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.
%D A160525 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160525 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160525 See Maple code for formula.
%e A160525 x^5+2*x^29+5*x^53+10*x^77+20*x^101+36*x^125+65*x^149+109*x^173+...
%p A160525 M1:=1200:
%p A160525 fm:=mul(1-x^n,n=1..M1):
%p A160525 A:=x^(1/7)*subs(x=x^(24/7),fm):
%p A160525 B:=x*subs(x=x^24,fm):
%p A160525 C:=x^7*subs(x=x^168,fm):
%p A160525 t1:=C/B^2;
%p A160525 t2:=series(t1,x,M1);
%p A160525 t3:=subs(x=y^(1/24),t2/x^5);
%p A160525 t4:=series(t3,y,M1/24);
%p A160525 t5:=seriestolist(t4); # A160525
%K A160525 nonn,new
%O A160525 0,2
%A A160525 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160524
%S A160524 8,15,17,37,41,46,51,53,55,65,75,77,102,106,110,116
%N A160524 Exceptional class of numbers n such that p(5n+4) == 0 mod 25, where p() = A000041().
%C A160524 The unexceptional class consists of the numbers n == 4 mod 5.
%D A160524 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160524 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%K A160524 nonn,new
%O A160524 1,1
%A A160524 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160521
%S A160521 1,8,44,192,726,2457,7648,22220,60993,159478,399906,966600,2261630,5139897,
%T A160521 11378988,24598683,52033372,107890610,219630050,439535138,865784403,1680352500,
%U A160521 3216454360,6077280123,11343018559,20928404349,38194869384,68989715838
%N A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.
%D A160521 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160521 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160521 See Maple code in A160458 for formula.
%e A160521 x^27+8*x^51+44*x^75+192*x^99+726*x^123+2457*x^147+7648*x^171+...
%K A160521 nonn,new
%O A160521 0,2
%A A160521 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160507
%S A160507 1,6,27,98,315,913,2462,6237,15035,34705,77231,166364,348326,710869,1417900,
%T A160507 2769730,5308732,9999185,18533944,33845975,60960273,108389248,190410133,
%U A160507 330733733,568388100,967054374,1629808139,2722189979,4508130889,7405471040
%N A160507 Coefficients in the expansion of C^5/B^6, in Watson's notation of page 106.
%D A160507 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160507 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160507 See Maple code in A160458 for formula.
%e A160507 x^19+6*x^43+27*x^67+98*x^91+315*x^115+913*x^139+2462*x^163+...
%K A160507 nonn,new
%O A160507 0,2
%A A160507 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160506
%S A160506 1,5,20,65,190,502,1245,2910,6505,13965,29005,58455,114810,220240,413775,
%T A160506 762635,1381550,2463060,4327445,7500260,12836645,21712470,36323930,60143320,
%U A160506 98620425,160238035,258110955,412367705,653709340,1028658150,1607306688
%N A160506 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 106.
%D A160506 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160506 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160506 See Maple code in A160458 for formula.
%e A160506 x^15+5*x^39+20*x^63+65*x^87+190*x^111+502*x^135+1245*x^159+...
%K A160506 nonn,new
%O A160506 0,2
%A A160506 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160463
%S A160463 1,4,14,40,105,249,562,1198,2460,4865,9352,17486,31973,57220,100550,173665,
%T A160463 295413,495339,819900,1340655,2167825,3468579,5495908,8628080,13428945,
%U A160463 20730689,31757174,48293585,72933885,109421095,163135433,241763735,356246552
%N A160463 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 106.
%D A160463 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160463 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160463 See Maple code in A160458 for formula.
%e A160463 x^11+4*x^35+14*x^59+40*x^83+105*x^107+249*x^131+562*x^155+...
%K A160463 nonn,new
%O A160463 0,2
%A A160463 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160462
%S A160462 1,3,9,22,51,106,215,411,766,1377,2423,4154,7001,11567,18834,30195,47809,
%T A160462 74735,115585,176847,268064,402598,599695,886116,1299808,1893115,2739248,
%U A160462 3938491,5629407,8000431,11309295,15904003,22256183,30998479,42981170,59337604
%N A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.
%D A160462 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160462 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160462 See Maple code in A160458 for formula.
%e A160462 x^7+3*x^31+9*x^55+22*x^79+51*x^103+106*x^127+215*x^151+...
%K A160462 nonn,new
%O A160462 0,2
%A A160462 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160461
%S A160461 1,2,5,10,20,35,63,105,175,280,444,685,1050,1575,2345,3439,5005,7195,10275,
%T A160461 14525,20405,28428,39375,54150,74080,100715,136265,183365,245645,327485,
%U A160461 434810,574790,756965,992950,1297940,1690500,2194642,2839695,3663225,4711160
%N A160461 Coefficients in the expansion of C/B^2, in Watson's notation of page 106.
%D A160461 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160461 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160461 See Maple code in A160458 for formula.
%e A160461 x^3+2*x^27+5*x^51+10*x^75+20*x^99+35*x^123+63*x^147+...
%K A160461 nonn,new
%O A160461 0,2
%A A160461 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160460
%S A160460 1,7,35,140,490,1541,4480,12195,31465,77525,183626,420077,932030,2011905,
%T A160460 4237130,8725671,17605602,34861815,67848095,129946805,245203642,456303872,
%U A160460 838178470,1520969100,2728472695,4841909821,8504898720,14794863270,25500965320
%N A160460 Coefficients in the expansion of C^6/B^7, in Watson's notation of page 106.
%D A160460 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A160460 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160460 See Maple code in A160458 for formula.
%e A160460 x^23+7*x^47+35*x^71+140*x^95+490*x^119+1541*x^143+...
%K A160460 nonn,new
%O A160460 0,2
%A A160460 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160459
%S A160459 2,13,66,286,1102,3879,12688,39050,114114,318863,856654,2222688,5589916,
%T A160459 13668072,32576016,75845402,172830788,386088741,846744800,1825447086,3872819904,
%U A160459 8094022001,16679126516,33916289400,68106769602,135148379654,265177195950
%N A160459 Omit first term of A160458 and divide by 5.
%D A160459 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression C^2/B^10.
%H A160459 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%K A160459 nonn,new
%O A160459 1,2
%A A160459 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160458
%S A160458 1,10,65,330,1430,5510,19395,63440,195250,570570,1594315,4283270,11113440,
%T A160458 27949580,68340360,162880080,379227010,864153940,1930443705,4233724000,
%U A160458 9127235430,19364099520,40470110005,83395632580,169581447000,340533848010
%N A160458 Coefficients in the expansion of C^2/B^10, in Watson's notation of page 106.
%D A160458 Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression C^2/B^10.
%H A160458 GDZ, <a href="http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618">Digitized volumes of Crelle</a>
%F A160458 See Maple code for formula.
%e A160458 1+10*x^24+65*x^48+330*x^72+1430*x^96+5510*x^120+19395*x^144+...
%p A160458 read format;
%p A160458 M1:=1200:
%p A160458 fm:=mul(1-x^n,n=1..M1):
%p A160458 A:=x^(1/5)*subs(x=x^(24/5),fm):
%p A160458 B:=x*subs(x=x^24,fm):
%p A160458 C:=x^5*subs(x=x^120,fm):
%p A160458 t1:=C^2/B^10;
%p A160458 t2:=series(t1,x,M1);
%p A160458 t3:=subs(x=y^(1/24),t2);
%p A160458 t4:=series(t3,y,M1/24);
%p A160458 t5:=seriestolist(t4); # A160458
%Y A160458 Cf. A160459.
%K A160458 nonn,new
%O A160458 0,2
%A A160458 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A160449
%S A160449 1,1,1,1,2,1,1,3,4,1,1,5,11,8,1,1,7,43,49,16,1,1,11,161,681,251,32,
%T A160449 1,1,15,901,14721,14491,1393,64,1
%N A160449 Array read by antidiagonals, giving number of isomorphism classes of multiple coverings of graphs with specified Betti number. See reference for precise definition.
%D A160449 J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See Table 2.
%Y A160449 Rows: A000012, A000079, A074528, A160450, A160454.
%Y A160449 Columns: A000041, A110143, A152612, A160446, A160447, A160448.
%Y A160449 Cf. A057004.
%K A160449 nonn,tabl,new
%O A160449 0,5
%A A160449 N. J. A. Sloane (njas(AT)research.att.com), Nov 13 2009

%I A167838
%S A167838 2,211,293,631,787,797,839,1249,1259,1399,1409,1471,1511,1637,1709,1801,
%T A167838 1811,1831,1847,2039,2053,2099,2179,2503,2521,2579,2633,2767,2777,2819,
%U A167838 2927,2939,3109,3137,3271,3433,3571,3593,3659,3779,3833,3863,3967,3989
%N A167838 Four-times-isolated primes: primes p such that neither p+-2, p+-4, p+-6 nor p+-8 is prime.
%C A167838 2 together with prime numbers, isolated from neighboring primes by>8.
%e A167838 a(1)=2 (-6,-4,-2,0,4,6,8,10 are nonprimes); a(2)=211 (203,205,207,209,213,215,217,219 are nonprimes).
%Y A167838 Cf. A137871, A167771.
%K A167838 nonn,new
%O A167838 1,1
%A A167838 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 13 2009

%I A167837
%S A167837 4,3,2,5,4,8,6,5,2,3,5,1,1,2,4,2,2,2,3,4,2,3,2,4,3,6,5,3,4,3,2,4,4,4,5,
%T A167837 7,8,3,1,5,5,5
%N A167837 Length of sections with distinct digits in decimal expansion of e (A001113).
%Y A167837 Cf. A104807.
%K A167837 nonn,new
%O A167837 1,1
%A A167837 Jani Melik (jani.melik(AT)gmail.com), Nov 13 2009

%I A167836
%S A167836 2718,281,82,84590,4523,53602874,713526,62497,75,724,70936,9,9,95,9574,
%T A167836 96,69,67,627,7240,76,630,35,3547,594,571382,17852,516,6427,427,46,6391,
%U A167836 9320,3059,92181,7413596,62904357,290,3,34295,26059,56307
%N A167836 Numbers with distinct digits appearing in partition of decimal expansion of e (A001113).
%C A167836 Start with decimal expansion of e: 2.71828182845904523536028747135266249775724709369995957496696... Part the sequence to the sections with distinct digits: s={2,7,1,8},{2,8,1},{8,2},{8,4,5,9,0},{4,5,2,3},{5,3,6,0,2,8,7,4}, {7,1,3,5,2,6},... Numbers from digits of s(n), leaving leading zeros: 2718,281,82,84590,4523,53602874,713526,... Leaving leading zeros as at first: a(34) = 3059 from {0,3,0,5,9}, and a(39) = 3 from {0,3}, ...
%Y A167836 Cf. A104819.
%K A167836 nonn,new
%O A167836 1,1
%A A167836 Jani Melik (jani.melik(AT)gmail.com), Nov 13 2009

%I A167833
%S A167833 2,211,293,409,479,631,691,701,709,719,787,797,839,919,929,1163,1171,
%T A167833 1201,1249,1259,1381,1399,1409,1471,1511,1523,1531,1637,1709,1733,1801,
%U A167833 1811,1823,1831,1847,1889,2039,2053,2099,2153,2161,2179,2221,2251,2459
%N A167833 Three-times-isolated primes: primes p such that neither p+-2, p+-4 nor p+-6 is prime.
%C A167833 2 together with prime numbers, isolated from neighboring primes by>6.
%e A167833 a(1)=2 (-4,-2,0,4,6,8 are nonprimes); a(2)=211 (205,207,209,213,215,217 are nonprimes).
%Y A167833 Cf. A137870, A167771.
%K A167833 nonn,new
%O A167833 1,1
%A A167833 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 13 2009

%I A167834
%S A167834 14,142,13562,37,3095,48,8016,8,8724,2096,9807,8569,6718,753,769480,731,
%T A167834 76,679,73,79,907324,7846210,7038,8503,87534,3276415,72,73501,38462,
%U A167834 30912,2970,249,2483605,58,5073,721,264,412,149709
%N A167834 Numbers with distinct digits appearing in partition of decimal expansion of square root of 2. (A002193)
%C A167834 Start with decimal expansion of sqrt(2): 1.41421356237309504880168872420969807856967187537694807317667... Part the sequence to the sections with distinct digits: S={1,4},{1,4,2},{1,3,5,6,2},{3,7},{3,0,9,5},{0,4,8},{8,0,1,6},... Numbers from digits of s(n), leaving leading zeros: 14,142,13562,37,3095,48,8016,...
%Y A167834 Cf. A104819.
%K A167834 nonn,new
%O A167834 1,1
%A A167834 Jani Melik (jani.melik(AT)gmail.com), Nov 13 2009

%I A167835
%S A167835 2,3,5,2,4,2,4,1,4,4,4,4,4,3,6,3,2,3,2,2,6,7,4,4,5,7,2,5,5,5,4,3,7,2,4,
%T A167835 3,3,3,6
%N A167835 Length of sections with distinct digits in decimal expansion of square root of 2. (A002193)
%Y A167835 Cf. A104807, A167834.
%K A167835 nonn,new
%O A167835 1,1
%A A167835 Jani Melik (jani.melik(AT)gmail.com), Nov 13 2009

%I A167826
%S A167826 0,0,0,0,0,2,8,26,74,194,482,1152,2674,6068,13524,29704,64460,138482,
%T A167826 294988,623834,1311086,2740666,5702270,11815752,24395678,50209572,
%U A167826 103048168,210965064,430938832,878534170
%N A167826 a(n) is the number of n-tosses having a run of 3 or more heads and a run of 3 or more tails for a fair coin.
%F A167826 a(n)=2^n-2*(Tribonacci(n+3)-Fibonacci(n+1))
%t A167826 b[1]=0;b[2]=1;b[3]=1;b[n_]:=b[n-1]+b[n-2]+b[n-3];Table[2^n-2*(Sum[b[n+3-i],{i,1,3}]-Fibonacci[n+1]),{n,1,30}]
%Y A167826 Cf. A167821, A050231, A008466,
%K A167826 easy,nonn,new
%O A167826 1,6
%A A167826 Veikko Pohjola (veikko(AT)nordem.fi), Nov 13 2009

%I A175040
%S A175040 0,1,8,3,256,5,46656,7,512,9,10000000,11,12,2197,14,170859375,4096,
%T A175040 289,104976,361,20,194481,10648,279841,7962624,15625,11881376,19683,
%U A175040 28,29,810000,961,32,33,70188843638032384,52521875,1679616,50653,38,1521
%N A175040 a(n) = Product of similar consecutive values of A072000(n).
%Y A175040 Cf. A072000.
%K A175040 easy,nonn,new
%O A175040 1,5
%A A175040 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Nov 13 2009
%E A175040 Corrected a(1) = 0, a(28) = 28, a(29) = 29, Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Nov 14 2009

%I A167824
%S A167824 24,8175,2779476,945013665,321301866624,109241689638495,
%T A167824 37141853175221676,12628120837885731345,4293523943027973435624,
%U A167824 1459785512508673082380815,496322780729005820036041476
%N A167824 Subsequence of A167709 whose indices are 3 mod 5 i.e. a(n)=A167709(5*n+3).
%F A167824 Recurrence formulae: a(n+2)=340*a(n+1)-a(n) or a(n+1)=170*a(n)+39*sqrt(19*(a(n))^2+81). G.f f(z)=(24+8175*z-24*340*z)/(1-340*z+z^2). a(n)=(105*sqrt(19)+456)/38*(170+39*sqrt(19))^n+(-105*sqrt(19)+456)/38*(170-39*sqrt(19))^n.
%e A167824 a(0)=A167709(3)=24, a(1)=A167709(8)=8175.
%p A167824 w(0):=24:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((105*sqrt(19)+456)/38*(170+39*sqrt(19))^(n)+(-105*sqrt(19)+456)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((24+8175*z-24*340*z)/(1-340*z+z^2)),z=0,21);
%K A167824 easy,nonn,new
%O A167824 0,1
%A A167824 Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 13 2009

%I A167825
%S A167825 220,74801,25432120,8646845999,2939902207540,999558103717601,
%T A167825 339846815361776800,115546917664900394399,39285612159250772318860,
%U A167825 13356992587227597688018001,4541338194045223963153801480
%N A167825 Subsequence of A167709 whose indices are 4 mod 5 i.e. a(n)=A167709(5*n+4).
%F A167825 Recurrence formulae: a(n+2)=340*a(n+1)-a(n) or a(n+1)=170*a(n)+39*sqrt(19*(w(n))^2+81). G.f f(z)=(220+74801*z-220*340*z)/(1-340*z+z^2). a(n)=(959*sqrt(19)+4180)/38*(170+39*sqrt(19))^n+(-959*sqrt(19)+4180)/38*(170-39*sqrt(19))^n.
%e A167825 a(0)=A167709(4)=220, a(1)=A167709(9)=74801.
%p A167825 w(0):=220:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((959*sqrt(19)+4180)/38*(170+39*sqrt(19))^(n)+(-959*sqrt(19)+4180)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((220+74801*z-220*340*z)/(1-340*z+z^2)),z=0,21);
%K A167825 easy,nonn,new
%O A167825 0,1
%A A167825 Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 13 2009

%I A175038
%S A175038 0,1,1,4,2,6,2,6,10,2,10,6,2,6,10,10,2,10,6,2,10,6,10,14,7,3,9,3,9,39,9,
%T A175038 15,3,27,3,15,15,9,15,15,3,27,3,9,3,33,33,9,3,9,15,3,27,15,15,15,3,15,9,
%U A175038 3,27,39,9,3,9,39,15,27,3,9,15,21,15,15,9,15,21,9,21,27,3,27,3,15,9,15
%N A175038 In the sequence of natural integers A000027, number of digits between successive primes.
%t A175038 Table[Length[Flatten[IntegerDigits/@Range[Prime[n]+1,Prime[n+1]-1]]],{n,200}]
%Y A175038 Cf. A000027, A113610.
%K A175038 base,nonn,new
%O A175038 0,4
%A A175038 Zak Seidov (zakseidov(AT)yahoo.com), Nov 13 2009

%I A167823
%S A167823 15,5124,1742145,592324176,201388477695,68471490092124,
%T A167823 23280105242844465,7915167311077025976,2691133605660945987375,
%U A167823 914977510757410558681524,311089662523913929005730785
%N A167823 Subsequence of A167709 whose indices are 2 mod 5 i.e. a(n)=A167709(5*n+2).
%F A167823 Recurrence formulae: a(n+2)=340*a(n+1)-a(n) or a(n+1)=170*a(n)+39*sqrt(19*(a(n))^2+81). G.f f(z)=(15+5124*z-15*340*z)/(1-340*z+z^2). a(n)=(66*sqrt(19)+285)/38*(170+39*sqrt(19))^n+(-66*sqrt(19)+285)/38*(170-39*sqrt(19))^n.
%e A167823 a(0)=A167709(2)=15, a(1)=A167709(7)=5124.
%p A167823 w(0):=15:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((66*sqrt(19)+285)/38*(170+39*sqrt(19))^(n)+(-66*sqrt(19)+285)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((15+5124*z-15*340*z)/(1-340*z+z^2)),z=0,21);
%K A167823 easy,nonn,new
%O A167823 0,1
%A A167823 Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 13 2009

%I A167821
%S A167821 0,0,2,6,16,38,86,188,402,846,1760,3630,7438,15164,30794,62342,125904,
%T A167821 253782,510758,1026684,2061730,4136990,8295872,16627166,33311646,
%U A167821 66716028,133582106,267406998,535206832,1071049286,2143127030
%N A167821 a(n) is the number of n-tosses having a run of 3 or more heads or a run of 3 or more tails for a fair coin (i.e. probability is a(n)/2^n).
%F A167821 G.f.: (2 x^2)/(1 - 3 x + x^2 + 2 x^3) a(n)=2^n - 2*Fibonacci(n+1)
%t A167821 CoefficientList[Series[(2 x^2)/(1 - 3 x + x^2 + 2 x^3), {x, 0, 30}], x] Table[2^n - 2*Fibonacci[n + 1], {n, 1, 31}]
%Y A167821 Cf. A050231, A008466
%K A167821 easy,nonn,new
%O A167821 1,3
%A A167821 Veikko Pohjola (veikko(AT)nordem.fi), Nov 13 2009

%I A167822
%S A167822 1,560,190399,64735100,22009743601,7483248089240,2544282340597999,
%T A167822 865048512555230420,294113949986437744801,99997877946876278001920,
%U A167822 33998984387987948082907999,11559554694037955471910717740
%N A167822 Subsequence of A167709 whose indices are 1 mod 5 i.e. a(n)=A167709(5*n+1).
%F A167822 Recurrence formulae: a(n+2)=340*a(n+1)-a(n) or a(n+1):=170*a(n)+39*sqrt(19*(a(n))^2+81). G.f f(z)=(1+560*z-1*340*z)/(1-340*z+z^2). a(n)=(10*sqrt(19)+19)/38*(170+39*sqrt(19))^n+(-10*sqrt(19)+19)/38*(170-39*sqrt(19))^n.
%e A167822 a(0)=A167709(1)=1, a(1)=A167709(6)=560.
%p A167822 w(0):=1:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((10*sqrt(19)+19)/38*(170+39*sqrt(19))^(n)+(-10*sqrt(19)+19)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((1+560*z-1*340*z)/(1-340*z+z^2)),z=0,21);
%K A167822 easy,nonn,new
%O A167822 0,2
%A A167822 Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 13 2009

%I A167820
%S A167820 0,351,119340,40575249,13795465320,4690417633551,1594728199942020,
%T A167820 542202897562653249,184347390443102162640,62677570547757172644351,
%U A167820 21310189638846995596916700,7245401799637430745779033649
%N A167820 Subsequence of A167709 whose indices are 0 mod 5 i.e. a(n)=A167709(5*n).
%F A167820 Recurrence formulae: a(n+2):=340*a(n+1)-a(n) or a(n+1):=170*a(n)+39*sqrt(19*(a(n))^2+81). G.f f(z)=(351*z)/(1-340*z+z^2) a(n)=(9*sqrt(19))/38*(170+39*sqrt(19))^n+(-9*sqrt(19))/38*(170-39*sqrt(19))^n
%e A167820 a(0)=0 because a(0)=A167709(0)=0, a(1)=351 because a(1)=A167709(5)=351.
%p A167820 w(0):=0:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((9*sqrt(19))/38*(170+39*sqrt(19))^(n)+(-9*sqrt(19))/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((351*z)/(1-340*z+z^2)),z=0,21);A2167709
%Y A167820 Cf. A167709
%K A167820 easy,nonn,new
%O A167820 0,2
%A A167820 Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 13 2009

%I A167817
%S A167817 1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,
%T A167817 3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,
%U A167817 3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1
%N A167817 Period 4: repeat 1 3 3 3.
%C A167817 Denominator of x(n)=x(n-1)+x(n-2), x(0)=1, x(1)=1/3; numerator=A167816.
%F A167817 a(n) = 3 - 2 * 0^(n mod 4).
%Y A167817 Cf. A130196.
%K A167817 frac,nonn,new
%O A167817 0,2
%A A167817 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 13 2009

%I A167819
%S A167819 9,10,12,14,16,17,18,20,22,23,24,25,27,31,37,39,41,43,49,53,54,62,67,71,
%T A167819 74,77,78,79,81,82,84,85,90,91,93,94,108,109,111,112,117,118,120,122,
%U A167819 124,125,130,131,133,134,148,149,151,152,157,158,160,161,162,164,168
%N A167819 Numbers with a distinct frequency for each ternary digit.
%C A167819 The smallest number in the sequence that actually contains all 3 ternary digits is 251 = 100022_3.
%e A167819 9 = 100_3 is in the sequence, as it has 2 0's, 1 1, and 0 2's. 1 is not in the sequence as it has the same number (0) of 0's and 2's.
%o A167819 (PARI) digits(n,b=10)=local(r);r=[];while(n>0,r=concat([n%b],r);n\=b);r
%o A167819 ina(n)=local(digs,v);digs=digits(n,3);v=vector(3);for(k=1,#digs,v[digs[k]+1]++);v[1]!=v[2]&v[1]!=v[3]&v[2]!=v[3]
%o A167819 for(n=1,250,if(ina(n),print1(n",")))
%Y A167819 Cf. A121977, A044951.
%K A167819 base,nonn,new
%O A167819 1,1
%A A167819 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 13 2009

%I A167816
%S A167816 0,1,1,2,1,5,8,13,7,34,55,89,48,233,377,610,329,1597,2584,4181,2255,
%T A167816 10946,17711,28657,15456,75025,121393,196418,105937,514229,832040,
%U A167816 1346269,726103,3524578,5702887,9227465,4976784,24157817,39088169
%N A167816 Numerator of x(n)=x(n-1)+x(n-2), x(0)=1, x(1)=1/3; denominator=A167817.
%C A167816 a(4*n) = A004187(n) = (a(4*n-1) + a(4*n-2))/3;
%C A167816 a(4*n+1) = A033889(n) = 3*a(4*n-1) + a(4*n-2);
%C A167816 a(4*n+2) = A033890(n) = a(4*n-1) + 3*a(4*n-2);
%C A167816 a(4*n+3) = A033891(n) = a(4*n-1) + a(4*n-2).
%H A167816 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%F A167816 a(n)=(a(n-1)*A093148(n+2)+a(n-2)*A093148(n+1))/A093148(n-1) for n>1.
%Y A167816 Cf. A000045, A167808.
%K A167816 frac,nonn,new
%O A167816 0,4
%A A167816 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 13 2009

%I A167810
%S A167810 1,3,13,86,760,8518,116278,1911198,37063964,835779524,21626042510,
%T A167810 635611172160
%N A167810 Number of admissible basis in the postage stamp problem for n denominations and h = 3 stamps.
%C A167810 A basis 1 = b_1 < b_2 ... < b_n is admissible if all the values 1 <= x <= b_n is obtainable as a sum of at most h (not necessarily distinct) numbers in the basis.
%D A167810 R. K. Guy, Unsolved Problems in Number Theory, C12.
%H A167810 R. Alter and J. A. Barnett, <a href="http://www.jstor.org/stable/2321610">A postage stamp problem</a>, Amer. Math. Monthly, 87 (1980), 206-210.
%H A167810 M. F. Challis, <a href="http://dx.doi.org/10.1093/comjnl/36.2.117">Two new techniques for computing extremal h-bases A_k</a>, Comp J 36(2) (1993) 117-126
%H A167810 Erich Friedman, <a href="http://www.stetson.edu/%7Eefriedma/mathmagic/0403.html">Postage stamp problem</a>
%H A167810 W. F. Lunnon, <a href="http://comjnl.oxfordjournals.org/cgi/content/abstract/12/4/377">A postage stamp problem</a>, Comput. J. 12 (1969) 377-380.
%H A167810 S. Mossige, <a href="http://www.jstor.org/stable/2007661">Algorithms for Computing the h-Range of the Postage Stamp Problem</a>, Math. Comp. 36 (1981) 575-582
%Y A167810 Other enumerations with different paramete