Primes that produce decreasing values of S000621(k)/prime(k).
2, 3, 5, 13, 23, 7, 41, 17, 37, 107, 149, 103, 31, 283, 739, 61, 919, 421, 223, 379, 307, 479, 389, 89, 83, 53, 47, 211, 1031, 1301, 659, 137, 1361, 373, 701, 157, 991, 29, 1171, 571, 521, 353, 263, 647, 1409, 251, 1201, 3169, 2269, 709
1
A link below shows a graph of S000621(n)/prime(n).
T. D. Noe, Plot of 50 terms
T. D. Noe, Plot of S000621(n)/prime(n)
Kevin A. Broughan and A. Ross Barnett, On the missing values of n! mod p, J. Ramanujan Math. Soc., 24(3):277–284, 2009.
Oleksiy Klurman and Marc Munsch, Distribution of factorials modulo p, arXiv 1505.01198 (May 05 2015)
(Mma) nn = 1000; f = Range[Prime[nn]]!; frac = Table[Length[Union[Mod[Take[f, p], p]]]/p, {p, Prime[Range[nn]]}]; t = Denominator /@ Reverse[Sort[Select[frac, # > 0.65 &]]]; t[[1]] = 2; t[[2]] = 3; t
Cf. A210184, S000620, S000621.
nonn
T. D. Noe, May 07 2015