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%I #23 Mar 06 2022 23:04:50
%S 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,
%T 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,
%U 6,5,5,3,4,5,7,3,5,4,2,4,4,4,4,6,2,3,4
%N Number of prime divisors of n! - 1 (counted with multiplicity).
%C The series is related to the product of primes and the "proof" of the existence of infinite many prime twins.
%H Amiram Eldar, <a href="/A054991/b054991.txt">Table of n, a(n) for n = 1..135</a>
%H R. G. Wilson v, <a href="/A038507/a038507.txt">Explicit factorizations</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha105.htm">Factorizations of many number sequences</a>
%e a(2)=0 because 2! - 1 = 1 (and this is not a prime number) a(5)=2 because 5! -1 = 119 = 7 * 17
%t a[q_] := Module[{x, n}, x=FactorInteger[q!-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
%t A054991[n_Integer] := PrimeOmega[n! - 1]; A054991[1] = 0; Table[A054991[n], {n,2,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)
%Y Cf. A033312, A054988, A054989, A054990, A054992.
%K nonn
%O 1,5
%A Arne Ring (arne.ring(AT)epost.de), May 30 2000
%E More terms from _Robert G. Wilson v_, Mar 24 2001
%E More terms from _Amiram Eldar_, Oct 03 2019
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