A054991 Number of prime divisors of n! - 1 (counted with multiplicity).

0, 0, 1, 1, 2, 1, 1, 2, 3, 2, 4, 1, 2, 1, 5, 2, 3, 3, 3, 2, 4, 3, 2, 2, 3, 2, 2, 4, 5, 1, 3, 1, 1, 2, 3, 2, 5, 1, 4, 2, 4, 4, 7, 4, 5, 5, 2, 4, 3, 2, 5, 5, 4, 6, 6, 5, 6, 5, 2, 3, 4, 4, 5, 4, 6, 4, 7, 2, 6, 5, 5, 3, 4, 5, 7, 3, 5, 4, 2, 4, 4, 4, 4, 6, 2, 3, 4

Offset: 1

Author

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

Keywords

• nonn

Comments

The series is related to the product of primes and the "proof" of the existence of infinite many prime twins.

Examples

			a(2)=0 because 2! - 1 = 1 (and this is not a prime number) a(5)=2 because 5! -1 = 119 = 7 * 17
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..135
• R. G. Wilson v, Explicit factorizations
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences

Crossrefs

Cf. A033312, A054988, A054989, A054990, A054992.

Programs

Mathematica

a[q_] := Module[{x, n}, x=FactorInteger[q!-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054991[n_Integer] := PrimeOmega[n! - 1]; A054991[1] = 0; Table[A054991[n], {n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

Extensions

More terms from Robert G. Wilson v, Mar 24 2001

More terms from Amiram Eldar, Oct 03 2019