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

1,3

Comments

The length of row n in A001269.

Links

Table of n, a(n) for n = 1..1122

S. S. Wagstaff, Jr., The Cunningham Project

Formula

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

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

Example

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

Mathematica

a[q_] := Module[{x, n}, x=FactorInteger[2^n+1]; n=Length[x]; Sum[Table[x[i]][2]], {i, n}][j]], {j, n}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n, 200}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

Prog

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

Crossrefs

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).

Sequence in context: A335420 A241318 A276064 * A096495 A276062 A324386

Adjacent sequences: A054989 A054990 A054991 * A054993 A054994 A054995

Keyword

nonn

Author

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

Extensions

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(1122) in b-file from Max Alekseyev, Jul 15 2023

Status

approved