A086257 -id:A086257 - cp's OEIS frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 2, 3, 4, 1, 2, 4, 2, 2, 4, 3, 2, 3, 4, 4, 6, 2, 3, 6, 2, 2, 5, 4, 5, 4, 3, 4, 4, 2, 3, 6, 2, 3, 7, 5, 3, 3, 3, 7, 6, 3, 3, 6, 6, 3, 5, 3, 4, 4, 2, 5, 7, 2, 6, 6, 3, 4, 5, 7, 3, 5, 3, 5, 7, 4, 6, 10, 2, 3, 10, 5, 6, 5, 4, 5, 5, 4, 4, 11, 6, 2, 5, 4, 5, 3, 5, 6, 9, 6, 2, 9, 3

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The length of row n in A001269.

			a(3) = 2 because 2^3 + 1 = 9 = 3*3.

Max Alekseyev, Table of n, a(n) for n = 1..1128
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), this sequence (b=2).
Cf. A000051, A002586, A002587, A003260, A001222, A001269, A001348, A054988, A054989, A054990, A054991, A000978.
Cf. A046051 (number of prime factors of 2^n-1).
Cf. A086257 (number of primitive prime factors).

a[n_] := Module[{x=FactorInteger[2^n+1]}, Sum[x[[i]][[2]], {i, Length[x]}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n,103}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A046051(2n) - A046051(n). - T. D. Noe, Jun 18 2003

a(n) = A001222(A000051(n)). - Amiram Eldar, Oct 04 2019

Extended by Patrick De Geest, Oct 01 2000
Terms to a(500) in b-file from T. D. Noe, Nov 10 2007
Deleted duplicate (and broken) Wagstaff link. - N. J. A. Sloane, Jan 18 2019
a(500)-a(1062) in b-file from Amiram Eldar, Oct 04 2019
a(1063)-a(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025

A046799 Number of distinct prime factors of 2^n+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 1, 2, 4, 2, 2, 3, 3, 2, 3, 4, 4, 3, 2, 3, 5, 2, 2, 4, 4, 5, 4, 3, 4, 3, 2, 3, 6, 2, 3, 5, 5, 3, 3, 3, 5, 5, 3, 3, 6, 5, 3, 4, 3, 4, 4, 2, 5, 5, 2, 6, 6, 3, 3, 4, 6, 3, 5, 3, 5, 6, 4, 6, 9, 2, 3, 6, 5, 6, 5, 4, 5, 4, 4, 4, 10, 6, 2, 4, 4, 5, 3, 5, 6, 7, 6, 2, 9, 3, 2

Offset: 0

Labos Elemer

nonn

The length of row n in A060444.

			For n=7, 129 = 3.43 has 2 prime factors, so a(7) = 2.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe; terms 501..1062 from Amiram Eldar)

Cf. A000051, A001221, A060444, A086257 (number of primitive prime factors).

PrimeNu[1 + 2^#] & /@ Range[0, 104] (* Jayanta Basu, Jun 29 2013 *)

a(n) = omega(2^n+1); \\ Michel Marcus, Mar 25 2017

a(n) = A001221(A000051(n)). - Amiram Eldar, Oct 04 2019

A069061 Sum of divisors of 2^n+1.

Original entry on oeis.org

4, 6, 13, 18, 48, 84, 176, 258, 800, 1302, 2736, 4356, 10928, 20520, 51792, 65538, 174768, 351120, 699056, 1110276, 3100240, 5048232, 11184816, 17041416, 49012992, 82623888, 211053040, 284225796, 727960800, 1494039792, 2863311536, 4301668356, 12611914848, 20788904016

Offset: 1

Benoit Cloitre, Apr 04 2002

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128 (terms 1..1062 from Amiram Eldar)

Cf. A000051, A000203, A002185, A002587, A002589, A046798, A046799, A053285, A054992, A057957, A059886, A085029, A086257, A112092, A250291, A274906, A295501.

Row sums of A374237.

DivisorSigma[1, 2^Range[50] + 1] (* Paolo Xausa, Jul 05 2024 *)

a(n) = sigma(2^n+1); \\ Michel Marcus, Nov 24 2013

a(n) = sigma(2^n+1).

a(n) = A000203(A000051(n)). - Michel Marcus, Nov 24 2013

More terms from Amiram Eldar, Oct 04 2019

A002185 Smallest primitive factor of 2^(2n+1) + 1.

Original entry on oeis.org

3, 1, 11, 43, 19, 683, 2731, 331, 43691, 174763, 5419, 2796203, 251, 87211, 59, 715827883, 67, 281, 1777, 22366891, 83, 2932031007403, 18837001, 283, 4363953127297, 307, 107, 2971, 571, 2833, 768614336404564651, 77158673929, 131, 7327657, 139, 56409643, 1753

Offset: 0

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 0..568
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002589, A086257, A112927.

Terms a(25) onward from Max Alekseyev, Oct 16 2022

A086258 a(n) is the smallest k such that 2^k+1 has n primitive prime factors.

Original entry on oeis.org

0, 14, 26, 46, 83, 118, 309, 194, 414, 538, 786, 958

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n+1 is called primitive if it does not divide 2^r+1 for any rA086257 for the number of primitive prime factors in 2^n+1. It is known that a(8) = 194.

Next term is > 666. - David Wasserman, Feb 25 2005

			a(2) = 14 because 2^14+1 = 5*29*113 and 29 and 113 do not divide 2^r+1 for r < 14.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

J. Brillhart et al., Factorizations of b^n +- 1 Available on-line

Cf. A086252, A086257.

More terms from David Wasserman, Feb 25 2005
a(11) from D. S. McNeil, Dec 19 2010
a(12) from Amiram Eldar, Oct 12 2019

A276172 Number of primitive prime divisors of 3^n - 2^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 3, 4, 1, 3, 1, 1, 3, 3, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 3, 2, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 3, 2, 4, 1, 3, 4, 3, 2, 2, 5, 3

Offset: 1

Michel Lagneau, Aug 23 2016

nonn

A prime factor of 3^n - 2^n is called primitive if it does not divide 3^r - 2^r for any positive r=2, Zsigmondy's theorem says that there is at least one primitive prime factor except two cases:
(i)   2^6 - 1^6
(ii)  n=2 and a+b is a power of 2.

			a(7) = 2 because 3^7 - 2^7 = 2059 = 29*71 => 29 and 71 do not divide 3^r - 2^r  for r < 7:
3^1 - 2^1 = 1;
3^2 - 2^2 = 5;
3^3 - 2^3 = 19;
3^4 - 2^4 = 65 = 5*13;
3^5 - 2^5 = 211;
3^6 - 2^6 = 665 = 5*7*19.

Amiram Eldar, Table of n, a(n) for n = 1..568
Eric Weisstein's World of Mathematics, Zsigmondy Theorem.

Cf. A001047, A086251, A086257.

f:= n -> nops(select(p -> numtheory:-order(3/2 mod p, p) = n, numtheory:-factorset(3^n-2^n)));
map(f, [$1..100]); # Robert Israel, Sep 14 2016

nMax=100; pLst={}; Table[f=Transpose[FactorInteger[3^n-2^n]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 1, nMax}]

a(1) corrected by Robert Israel, Sep 14 2016

cp's OEIS Frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

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A046799 Number of distinct prime factors of 2^n+1.

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A069061 Sum of divisors of 2^n+1.

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A002185 Smallest primitive factor of 2^(2n+1) + 1.

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A086258 a(n) is the smallest k such that 2^k+1 has n primitive prime factors.

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A276172 Number of primitive prime divisors of 3^n - 2^n.

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