A066856 a(n) = omega(n!+1), where omega is the number of distinct primes dividing n, A001221.

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 5, 3, 6, 2, 2, 3, 3, 3, 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

Robert G. Wilson v, Jan 21 2002

Keywords

• hard
• nonn

Comments

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 09 2003

Links

• Amiram Eldar, Table of n, a(n) for n = 1..139
• William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
• Paul Leyland, Factors of n!+1 [Typo in URL corrected by R. J. Mathar, Nov 21 2008]
• Hisanori Mishima, Appendix 1. Factorization results for n!+1
• Hisanori Mishima, Bernoulli numbers (n = 2 to 114)

Crossrefs

Cf. A054990 (bigomega(n!+1)), A002981 (n!+1 is prime), A064237 (n!+1 divisible by a square), A084846 (mu(n!+1)).

Programs

Magma

[#PrimeDivisors(Factorial(n) + 1): n in [1..55]]; // Vincenzo Librandi, Oct 11 2018

Mathematica

Table[ Length[ FactorInteger[ n! + 1]], {n, 1, 15}]
PrimeNu[Range[50]! + 1] (* Paolo Xausa, Feb 07 2025 *)

PARI

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

Extensions

More terms from Rick L. Shepherd, Jun 09 2003