Least prime p such that there are n numbers m such that p divides m! + 1.
2, 7, 23, 59, 227, 401, 71, 3643, 62939, 480209, 292627, 1089427
1
Of course, all the m are less than p. The 14th term is 2374649. See S000684 for the triangle of numbers.
T. D. Noe, Plot of 12 terms
(Mma) nn = 8; t = Table[0, {nn}]; found = 0; p = 1; While[found < nn, p = NextPrime[p]; cnt = 0; f = 1; Do[f = Mod[f*m, p]; If[f + 1 == p, cnt++], {m, p - 1}]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = p; found++]], {p, found}]; t
nonn,hard
T. D. Noe, May 18 2015