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

Author

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

Keywords

• nonn

Comments

The length of row n in A001269.

Examples

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

Links

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

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).

Programs

Mathematica

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 *)

PARI

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

Formula

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

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

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(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025