A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 5, 3, 6, 2, 2, 3, 3, 4, 2, 2, 2, 1, 2, 3, 5, 4, 4, 5, 2, 5, 6, 1, 2, 4, 7, 1, 3, 4, 3, 3, 3, 4, 2, 5, 5, 6, 4, 4, 2, 2, 4, 3, 4, 2, 4, 4, 3, 5, 3, 4, 5, 4, 5, 6, 5, 2, 7, 1, 4, 2, 3, 1, 6, 3, 4, 7, 3, 3, 3, 5, 5, 4, 3, 8, 3, 6, 2, 4, 3, 4, 5, 6, 6, 5, 5, 4, 5

Offset: 1

Author

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

Keywords

• nonn
• hard

Comments

The smallest k! with n prime factors occurs for n in A060250.

103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd, Jun 10 2003

Examples

			a(2)=2 because 4! + 1 = 25 = 5*5
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..139
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• R. G. Wilson v, Explicit factorizations
• Paul Leyland, Factors of n!+1.

Crossrefs

Cf. A000040 (prime numbers), A001359 (twin primes).

Cf. A038507, A054988, A054989, A054991, A054992.

Cf. A066856 (number of distinct prime divisors of n!+1), A084846 (mu(n!+1)).

Programs

Mathematica

a[q_] := Module[{x, n}, x=FactorInteger[q!+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054990[n_Integer] := PrimeOmega[n! + 1]; Table[A054990[n], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

PARI

for(n=1,64,print1(bigomega(n!+1),","))

Extensions

More terms from Robert G. Wilson v, Mar 23 2001

More terms from Rick L. Shepherd, Jun 10 2003