## S000676

The composite number A256519(n) divides a(n)! + 1.

4, 5, 12, 18, 18, 18, 6, 16, 18, 18, 9, 18, 18, 18, 18, 23, 16, 16, 40, 36, 23, 18, 7, 9, 51, 16, 16, 63, 40, 22, 18, 61, 40, 16, 9, 102, 63, 18, 36, 15, 18, 18, 18, 63, 8, 18, 18, 210, 35, 51, 36, 36, 66, 51, 58, 198, 18, 225, 222, 18, 72, 78, 93, 18, 36, 89

1

As illustrated in sequences S000670 to S000675, for prime p, there can be many m such that p divides m! + 1. For composite numbers c, there appears to be at most one number m such that c divides m! + 1 and we find that m < sqrt(c). A PC program was used to generate numbers. Each number appears only a finite number of times.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Eric W. Weisstein, MathWorld: Wilson’s Theorem

(Mma) t2 = {}; Do[If[! PrimeQ[n], mx = FactorInteger[n][[1, 1]]; t = {}; f = 1; Do[f = Mod[k*f, n]; If[f + 1 == n, AppendTo[t, k]], {k, mx - 1}]; If[Length[t] > 0, AppendTo[t2, {n, t}]]], {n, 10^6}]; First /@Transpose[t2][[2]]

Cf. A256519,

nonn

T. D. Noe, Jun 12 2015