Search: id:A000961
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%I A000961 M0517 N0185
%S A000961 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,59,
%T A000961 61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,128,131,
%U A000961 137,139,149,151,157,163,167,169,173,179,181,191,193,197,199,211,223,227
%N A000961 Prime powers p^k (p prime, k >= 0).
%C A000961 Since 1 = p^0 does not have a well defined prime base p, it is sometimes 
               not regarded as a prime power.
%C A000961 These numbers are (apart from 1) the numbers of elements in finite fields. 
               - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Aug 11 2004
%C A000961 Numbers whose divisors form a geometrical progresion. The divisors of 
               p^k are 1, p, p^2, p^3, ...p^k. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Jan 09 2002
%C A000961 a(n) = A025473(n)^A025474(n). - David Wasserman (wasserma(AT)spawar.navy.mil), 
               Feb 16 2006
%C A000961 a(n) = A117331(A117333(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 08 2006
%C A000961 These are also precisely the orders of those finite affine planes that 
               are known to exist as of today. (The order of a finite affine plane 
               is the number of points in an arbitrarily chosen line of that plane. 
               This number is unique for all lines comprise the same number of points.) 
               - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006
%C A000961 Except for first term, the index of the second number divisible by n 
               in A002378, if the index equals n. - Mats Granvik (mgranvik(AT)abo.fi), 
               Nov 18 2007
%C A000961 These are precisely the numbers such that lcm(1,...,m-1)<lcm(1,...,m) 
               (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). 
               We have a(n+1)=a(n)+1 iff n is a Mersenne prime or n+1 is a Fermat 
               prime. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 18 2007
%C A000961 The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. 
               - Zak Seidov (zakseidov(AT)yahoo.com), Feb 06 2008
%C A000961 Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] 
               yields a prime at x=1, cf. A020500. - M. F. Hasler (www.univ-ag.fr/
               ~mhasler), Apr 04 2008
%C A000961 Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,
               22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly 
               the indices for which Phi[k](-1) is prime. - M. F. Hasler (www.univ-ag.fr/
               ~mhasler), Apr 04 2008
%C A000961 A143201(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 12 2008]
%C A000961 Number of distinct primes dividing n=omega(n)<2. [From Juri-Stepan Gerasimov 
               (2stepan(AT)rambler.ru), Oct 30 2009]
%D A000961 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 870.
%D A000961 M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993
%D A000961 R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their 
               Applications, Cambridge 1986, Theorem 2.5, p. 45.
%D A000961 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000961 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000961 T. D. Noe, <a href="b000961.txt">Table of n, a(n) for n = 1..10000</a>
%H A000961 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000961 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimePower.html">Link to a section of The World of Mathematics.</
               a>
%H A000961 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ProjectivePlane.html">Link to a section of The World of Mathematics.</
               a>
%H A000961 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000961 m=a(n) for some n <=> lcm(1,...,m-1)<lcm(1,...,m), where lcm(1...0):=0 
               as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) 
               for some k (by Catalan's conjecture). - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Jan 18 2007
%F A000961 A001221(a(n))<2. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Oct 30 2009]
%p A000961 readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then 
               printf(`%d,`,n) fi: od:
%t A000961 Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
%t A000961 Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]
%t A000961 max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], 
               {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 
               1000}]; a(*Artur Jasinski*)
%o A000961 (MAGMA) [ n : n in [1..1000] | IsPrimePower(n) ];
%o A000961 (PARI) A000961(n,l=-1,k=0)=until(n--<1,until(l<lcm(l,k++),); l=lcm(l,
               k));k print_A000961(lim=999,l=-1)=for(k=1,lim,l==lcm(l,k)&next;l=lcm(l,
               k);print1(k,",")) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Jan 18 2007
%o A000961 (PARI) isA000961(n) = (omega(n) == 1 | n == 1) [From Michael Porter (michael_b_porter(AT)yahoo.com), 
               Sep 23 2009]
%o A000961 (PARI) nextA000961(n) = {local(m,r,p);m=2*n;for(e=1,ceil(log(n+0.01)/
               log(2)),r=(n+0.01)^(1/e);p=prime(primepi(r)+1);m=min(m,p^e));m} [From 
               Michael Porter (michael_b_porter(AT)yahoo.com), Nov 02 2009]
%Y A000961 Cf. A010055, A065515, A095874, A025473.
%Y A000961 Cf. indices of record values of A003418; A000668 and A019434 give a member 
               of twin pairs a(n+1)=a(n)+1.
%Y A000961 A138929(n) = 2*a(n).
%Y A000961 Sequence in context: A059046 A144711 A036116 this_sequence A128603 A096165 
               A164336
%Y A000961 Adjacent sequences: A000958 A000959 A000960 this_sequence A000962 A000963 
               A000964
%K A000961 nonn,easy,core,nice,new
%O A000961 1,2
%A A000961 N. J. A. Sloane (njas(AT)research.att.com).

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