p - 1 - greatest number not in the irregular table of k! mod p for k = 1..p-1 for primes p.
1, 1, 1, 2, 5, 4, 2, 1, 2, 4, 3, 4, 3, 2, 2, 2, 1, 4, 1, 2, 2, 1, 1, 1, 11, 2, 1, 3, 5, 4, 5, 3, 3, 2, 1, 1, 2, 1, 4, 1, 9, 1, 2, 1, 3, 4, 1, 2, 1, 4, 4, 3, 1, 2, 1, 3, 2, 2, 3, 2, 3, 11, 2, 5, 10, 1, 1, 2, 1, 2, 3, 1, 12, 2, 1, 2, 1, 2, 2, 1, 1, 1, 7, 10, 2, 2
3
Note that p-1 is always in the table k! mod p.
T. D. Noe, Plot of terms 3..1000
T. D. Noe, Table of terms 3..1000
Tim Trudgian, There are no socialist primes less than 10^9, Integers 14 (2014), A63.
(Mma) nn = 100; f = Table[n!, {n, Prime[nn]}]; Table[s = Table[Mod[f[[n]], p], {n, p - 1}]; p - 1 - Complement[Range[p - 1], Union[s]][[-1]], {p, Prime[Range[3, nn]]}]
nonn
T. D. Noe, Jan 12 2015