This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A166864 #7 Feb 16 2025 08:33:11
%S A166864 17,23,29,31,53,59,61,67,71,73,83,89,97,103,107,109,137,139,149,151,
%T A166864 167,193,199,211,223,227,233,239,251,271,277,283,307,311,331,359,379,
%U A166864 389,397,401,419,431,439,449,457,461,463,467,479,487,499,503,521,547,557
%N A166864 Primes p that divide n! - 1 for some n > 1 other than p-2.
%C A166864 Since n! - 1 = 0 for n=1 and n=2, the restriction n > 1 needed to be placed.
%C A166864 For n >= p, p is one of the factors of n!, so p cannot divide n! - 1.
%C A166864 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.
%C A166864 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.
%H A166864 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WilsonsTheorem.html">Wilson's Theorem</a>
%e A166864 17 is included in the sequence since 17 divides 5! - 1 = 119.
%e A166864 19 is not included in the sequence since the only n for which 19 divides n! - 1 is n = 17.
%o A166864 (PARI) isA166864(n) = {local(r);r=0;for(i=2,n-3,if((i!-1)%n==0,r=1));r}
%Y A166864 Cf. A000142, A002582, A033312, A054415.
%K A166864 nonn
%O A166864 1,1
%A A166864 _Michael B. Porter_, Oct 22 2009