1,2

This sequence differs from A061955 in that all least significant zeros are removed before concatenation.

The next term is > 410000. - Larry Reeves, Jan 16, 2002

Any further terms are > 1125000. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 24 2009]

Index entries for related sequences

7654321 -> (111)(011)(101)(001)(11)(01)(1) base 2 -> 1111110111111 base 2 = 8127 and 7 divides 8127

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Sequence in context: A041663 A042825 A076033 this_sequence A045424 A106968 A079779

Adjacent sequences: A029516 A029517 A029518 this_sequence A029520 A029521 A029522

nonn,base,more

Olivier Gerard (olivier.gerard(AT)gmail.com)

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12, 2002

Additional comments and example from Larry Reeves (larryr(AT)acm.org), May 25 2001

Terms verified by Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 24 2009

1 more term from Sean A. Irvine (sairvin(AT)xtra.co.nz), Oct 04 2009

1,2

This sequence differs from A061978 in that all least significant zeros are removed before concatenation.

The next term is > 400000. - Larry Reeves, Jan 16, 2002

Index entries for related sequences

See A029519 for example.

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Sequence in context: A032640 A127174 A044055 this_sequence A014962 A014945 A045590

Adjacent sequences: A029539 A029540 A029541 this_sequence A029543 A029544 A029545

nonn,base,more

Olivier Gerard (olivier.gerard(AT)gmail.com)

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12, 2002

Additional comments and more terms from Larry Reeves (larryr(AT)acm.org), Jun 04 2001

1,2

This sequence differs from A029518 in that all least significant zeros are removed before concatenation.

The next term is > 400000. - Larry Reeves, Jan 16, 2002

Index entries for related sequences

See A061931 for example.

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Sequence in context: A029470 A161388 A029518 this_sequence A095039 A022940 A025207

Adjacent sequences: A061951 A061952 A061953 this_sequence A061955 A061956 A061957

nonn,base,more

Larry Reeves (larryr(AT)acm.org), May 24 2001

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12, 2002

1,1

Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.

Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24.

See also the Cohen-te Reile links under A019276.

The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Feb 29 2008

The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 11 2008

Largest proper divisor of A075398(n) if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Apr 25 2008

Indices of hexagonal numbers (A000384) that are also even perfect numbers, if there are no odd superperfect numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2008]

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Anonymous, Superperfect Numbers:Definition

Experimental Mathematics, Home Page

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos".

a(n)=(1 + A000668(n))/2, if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 11 2008

Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A075398(n)/2 = A032742(A075398(n)). - Omar E. Pol (info(AT)polprimos.com), Apr 25 2008

sigma(sigma(4))=2*4, so 4 is in the sequence.

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652.

Cf. A032742, A075398.

Cf. A000384. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2008]

Sequence in context: A154004 A060656 A061286 this_sequence A061652 A162119 A155519

Adjacent sequences: A019276 A019277 A019278 this_sequence A019280 A019281 A019282

nonn,more,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments and 2 more terms from Jud McCranie (j.mccranie(AT)comcast.net), Jun 01 2000

1,1

A prime p is in class 1 if (p+1)'s largest prime factor is 2 or 3. If (p+1) has other prime factors, p's class is one more than the largest class of its prime factors.

John W. Layman (layman(AT)math.vt.edu) observes that for n=10..13, the ratios r(n)= a(n)/a(n-1) are increasingly close to an integer, being 1.9999981, 7.99999906, 8.00000059 and 7.999999985.

2*a(15)-1 = 47738922361 < a(16) <= 429650301257 = 9*2*a(15)-1 - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

Layman's observation is a consequence of a(n+1) = m*a(n)-1 for (n,m)=(1,7),(3,2),(4,14),(9,2),(10,8),(12,8),(14,14), while a(12) = 8 a(11)+5 is a coincidence which does not fit into that scheme. This relationship is not unusual since any N+ prime p is by definition such that p+1 = m*q where q is a (N-1)+ prime and m = (p+1)/q must be even since p,q are odd (except for q=2, allowing the odd m=7 for n=1 above) and the least N+ prime has good chances of having q equal to the least (N-1)+ prime. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

a(16) <= 288310406533, with equality if 144155203267 is the 12th 15+ prime; a(17) <= 1833174628057, with equality if 916587314029 is the 10th 16+ prime; a(18) <= 3666349256113, with equality if a(17) = 1833174628057; a(19) <= 65994286610033, with equality if 41431295033731 is the third 18+ prime; a(20) <= 764276710625653, with equality if 382138355312827 is the third 19+ prime. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

a(16) calculated using A129475(n) up to n=19. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007

R. K. Guy, Unsolved Problems in Number Theory, A18.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

a(n+1) >= 2*a(n)-1 since a(n+1)+1 = p*q with p of class n+ (thus >= a(n) and odd) and thus q >= 2 (even and positive). a(n+1) <= min { p = 2*k*a(n)-1 | k=1,2,3,... such that p is prime }. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

1553 is in class 4 because 1553+1 = 2*3*7*37; 7 is in class 1 and 37 is in class 3. 37 is in class 3 because 37+1 = 2*19 and 19 is in class 2. 19 is in class 2 because 19+1 = 2*2*5 and 5 is in class 1. 5 is in class 1 because 5+1=2*3.

PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] + 1]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassPlusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 1, 28700000}]; a

(PARI) checkclass(n, p)={ n=factor(n+1)[, 1]; if( n[ #n] <= 3, return(1)); if( #p <= 1 | n[ #n] < p[ #p], return(2)); n[1]=p[ #p]; p=vecextract(p, "^-1"); forstep( i=#n, 2, -1, if( n[i] < n[1], break); if( checkclass(n[i], p) > #p, return(2+#p))); 0 }; A005113(n, p, a=[])={ while( #a<n, until( checkclass(p, a) > #a, p=nextprime(p+1)); a=concat(a, p); p=a[ #a]*2-2); a }; A005113(11) /* < 10 sec @ 2 GHz */ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

(PARI) class( n, s=+1 /* +1 for n+ class, -1 for n- class */ ) = { if( isprime(n), if(( n=factor(n+s)[, 1] ) & n[ #n]>3, vecsort(vector(#n, i, class(n[i], s)))[ #n]+1, 1), 0) }; someofnextclass( a, limit=0, s=0, b=[], p)={ if(!s, /* guess + or - */ s=( class(a[1]) & class(a[1])==class(a[2]) )*2-1 ); print("looking for primes of class ", 1+class( a[1], s), ["+", "-"][1+(s<0)] ); for( i=1, #a, p=-s; until( p>=limit, until( isprime(p), p+=a[i]<<1 ); b=concat(b, p); if( !limit, limit=p)) ); vecsort(b) }; c=A090468; for(i=15, 20, c=someofnextclass(c, 9e12); print("least prime of class ", i, "+ is <= ", c[1])) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

Cf. A056637, A005105, A005106, A005107, A005108, A019268.

Cf. A081633 - A081639, A084071, A090468, A129474 - A129476, A129469.

Sequence in context: A063092 A034011 A085497 this_sequence A072857 A119535 A011919

Adjacent sequences: A005110 A005111 A005112 this_sequence A005114 A005115 A005116

more,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Extended through a(12) by Robert G. Wilson v (rgwv(AT)rgwv.com).

a(13) from John W. Layman (layman(AT)math.vt.edu).

a(14) from Don Reble, Apr 11, 2003. 4294967296 < a(15) <= 23869461181.

a(15) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006

a(7) corrected by Tomas Oliveira e Silva, Oct 27 2006

a(16) from M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007

1,1

For primes of the form (9+k!)/9 see A139068

a(10) corrected from 1053 to 1056 by Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Jul 12 2008

No other n exists, for n<= 6000. - Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008

a(11) = 3562 because 3562 is the 11th natural number for which k!/9 + 1 is prime. 3562 is the new term.

a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, n]], {n, 1, 500}]; a

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072.

Sequence in context: A120044 A002686 A110348 this_sequence A034469 A134114 A071586

Adjacent sequences: A137387 A137388 A137389 this_sequence A137391 A137392 A137393

nonn,more

Artur Jasinski (grafix(AT)csl.pl), Apr 09 2008

Edited by N. J. A. Sloane (njas(AT)research.att.com), May 15 2008 at the suggestion of Richard J. Mathar

3562 from Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008

1,1

For the corresponding primes p see A139075.

a(9)>5000, a(13)>5000, a(22)>5000, a(23) = 1579. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 21 2008, Apr 22 2008

a(10)=5, a(11)=13, a(12)=5

a(14)= 17, a(15)=7, a(16)=13, a(17)=43, a(18)=7, a(19)=31, a(20)=5, a(21)=7

a(24)=7, a(25)=47, a(26)=17, a(27)=17, a(28)=7, a(29)=241, a(30)=5,

a(31)=61, a(32)=11, a(33)=17, a(34)=17, a(35)=29, a(36)=11, a(37)=61,

a(38)=103, a(39)=89, a(40)=7, a(41)=131, a(42)=11, a(43)=71, a(44)=13,

a(45)=7, a(46)=43, a(47)=73, a(48)=67, a(49)=347, a(50)=31, a(51)=19,

a(52)=17, a(53)=347, a(54)=11, a(55)=13, a(56)=13, a(57)=31, a(58)=73,

a(59)=641, a(60)=5

a(23) = 1579. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 11 2008.

Smallest daughter factorial prime p of order n, i.e. smallest prime of the form (p!+n)/n where p is prime.

For smallest mother factorial prime p of order n see A139075

For smallest father factorial prime p of order n see A139207

For smallest son factorial prime p of order n see A139206

a(1) = 2 because 2 is the first prime and 2!/1 + 1 = 3 is prime

a(2) = 2 because 2 is the first prime and 2!/2 + 1 = 2 is prime

a(3) = 3 because 3!/3 + 1 = 3 is prime

a = {}; Do[k = 1; While[ ! PrimeQ[(Prime[k]! + n)/n], k++ ]; AppendTo[a, Prime[k]], {n, 1, 8}]; a

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072, A139073, A139074, A139074, A139075, A136019, A136020, A136026, A136027.

Sequence in context: A032157 A153926 A035425 this_sequence A035428 A100142 A079953

Adjacent sequences: A139071 A139072 A139073 this_sequence A139075 A139076 A139077

more,nonn

Artur Jasinski (grafix(AT)csl.pl), Apr 08 2008, Apr 21 2008

1,1

Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 09 2003

The squares of these numbers are Fermat pseudoprimes to base 2 (A001567). - T. D. Noe (noe(AT)sspectra.com), May 22 2003

Primes p that divide the numerator of the harmonic number H((p-1)/2); that is, p divides A001008((p-1)/2). - T. D. Noe (noe(AT)sspectra.com), Mar 31 2004

In a 1977 paper, Wells Johnson, citing a suggestion from Lawrence Washington, pointed out the repetitions in the binary representations of the numbers which are one less than the two known Wieferich primes; i.e. 1092 = 10001000100 (base 2); 3510 = 110110110110 (base 2). It is perhaps worth remarking that 1092 = 444 (base 16) and 3510 = 6666 (base 8), so that these numbers are small multiples of repunits in the respective bases. Whether this is mathematically significant does not appear to be known. - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Sep 29 2007

A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 09 2008, Aug 24 2008

Dorais and Klyve (see reference) reported on November 27, 2008, that there are no other Wieferich primes up to 6.7*10^15. [From Peter Luschny (peter(AT)luschny.de), Feb 10 2009]

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.

R. K. Guy, Unsolved Problems in Number Theory, A3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 91.

Y. Hellegouarch, "Invitation aux mathematiques de Fermat Wiles", Dunod, 2eme Edition, pp. 340-341.

J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp., 75 (2005), 1559-1563.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 263.

J. Silverman, "Wieferich's Criterion and the abc Conjecture", J. Number Th. 30 (1988) 226-237.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 163.

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv.org/abs/0806.3412

Joerg Arndt, Fxtbook

C. K. Caldwell, The Prime Glossary, Wieferich prime

C. K. Caldwell, Prime-square Mersenne divisors are Wieferich

D. X. Charles, On Wieferich Primes

R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.

J. K. Crump, Joe's Number Theory Web, Weiferich Primes

John Blythe Dobson, A note on the two known Wieferich Primes

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)

W. Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die Reine und Angewandte Mathematik 292 (1977): 196-200.

C. McLeman, PlanetMath.org, Wieferich prime

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Wieferich Home Page, Search for Wieferich primes

Wikipedia, Wieferich prime

P. Zimmermann, RECORDS FOR PRIME NUMBERS

F.G. Dorais and D.W. Klyve, Near Wieferich Primes up to 6.7*10^15, November 27, 2008, PDF [From Peter Luschny (peter(AT)luschny.de), Feb 10 2009]

wieferich := proc (n) local nsq, remain, bin, char: if (not isprime(n)) then RETURN("not prime") fi: nsq := n^2: remain := 2: bin := convert(convert(n-1, binary), string): remain := (remain * 2) mod nsq: bin := substring(bin, 2..length(bin)): while (length(bin) > 1) do: char := substring(bin, 1..1): if char = "1"

then remain := (remain * 2) mod nsq fi: remain := (remain^2) mod nsq: bin := substring(bin, 2..length(bin)): od: if (bin = "1") then remain := (remain * 2) mod nsq fi: if remain = 1 then RETURN ("Wieferich prime") fi: RETURN ("non-Wieferich prime"): end: # from UlrSchimke(AT)aol.com, Nov 01, 2001

Select[Prime[Range[10^3*5]], Round[(2^(#-1)-1)/#^2]==((2^(#-1)-1)/#^2) &] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), May 01 2008)

See A007540 for a similar problem. Cf. A001567, A077816.

Sequence in context: A023698 A038469 A077816 this_sequence A115192 A091674 A022197

Adjacent sequences: A001217 A001218 A001219 this_sequence A001221 A001222 A001223

nonn,hard,bref,nice,more

N. J. A. Sloane (njas(AT)research.att.com).

Sequence is believed to be infinite, although there are no other terms < 4*10^12.

Wieferich Home Page link from Filip Zaludek (filip.zaludek(AT)gtsnovera.cz), Feb 05 2008

1,1

Next term, if it exists, is > 10^125074. - David Wasserman (wasserma(AT)spawar.navy.mil), Aug 13 2004

Since x+y is a factor of x^m+y^m if m is odd, 2^m+3^m is divisible by 2+3=5 unless m is zero or a power of 2. This is similar to Fermat numbers 1+2^m. - Michael Somos, Aug 27 2004

m=0: 1+1, m=1: 2+3, m=2: 4+9, m=4: 16+81

a={}; Do[If[PrimeQ[p=2^n+3^n], AppendTo[a, p]], {n, 0, 10^3}]; a [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 07 2008]

Cf. A094474-A094499.

Sequence in context: A075742 A075737 A100843 this_sequence A158712 A090472 A120266

Adjacent sequences: A082098 A082099 A082100 this_sequence A082102 A082103 A082104

more,nonn

Labos E. (labos(AT)ana.sote.hu), Apr 14 2003

1,1

All a(n) are primes. (3^n + 5^n)/8 = A074606[n]/8 = A081186[n]/4. Corresponding primes of the form (3^n + 5^n)/2^3 are listed in A121938[n] = A079773[a(n)] = {19,421,10039,95383574161,2384331073699,1925929944387235853055979210606894889560480247048440342330377620014353281101,...}.

Do[f=5^n+3^n; If[PrimeQ[f/2^3], Print[{n, f/2^3}]], {n, 1, 1231}]

Cf. A074606, A081186, A121824, A121877, A005058, A005059, A121938, A109347, A079773.

Sequence in context: A087126 A062547 A125739 this_sequence A137258 A053341 A086086

Adjacent sequences: A122850 A122851 A122852 this_sequence A122854 A122855 A122856

more,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Sep 14 2006