A166864 Primes p that divide n! - 1 for some n > 1 other than p-2.

17, 23, 29, 31, 53, 59, 61, 67, 71, 73, 83, 89, 97, 103, 107, 109, 137, 139, 149, 151, 167, 193, 199, 211, 223, 227, 233, 239, 251, 271, 277, 283, 307, 311, 331, 359, 379, 389, 397, 401, 419, 431, 439, 449, 457, 461, 463, 467, 479, 487, 499, 503, 521, 547, 557

Offset: 1

Author

Michael B. Porter, Oct 22 2009

Keywords

• nonn

Comments

Since n! - 1 = 0 for n=1 and n=2, the restriction n > 1 needed to be placed.

For n >= p, p is one of the factors of n!, so p cannot divide n! - 1.

For n = p-1, by Wilson's Theorem, (p-1)! = -1 (mod p), so p divides (p-1)! + 1, and cannot also divide (p-1)! - 1 unless p = 2.

For n = p-2, again by Wilson's Theorem, (p-1)! = (p-1)(p-2)! = (-1)(p-2)! = -1 (mod p), so (p-2)! = 1 (mod p) and p divides (p-2)! - 1. As a result, only 2 <= n <= p-3 needs to be searched.

Examples

			17 is included in the sequence since 17 divides 5! - 1 = 119.
19 is not included in the sequence since the only n for which 19 divides n! - 1 is n = 17.
		

Links

• Eric Weisstein's World of Mathematics, Wilson's Theorem

Crossrefs

Cf. A000142, A002582, A033312, A054415.

Programs

PARI

isA166864(n) = {local(r);r=0;for(i=2,n-3,if((i!-1)%n==0,r=1));r}