%I #18 Jan 09 2024 18:20:41
%S 1,2,24,34,494,675,4419,4008,4944,13136,21730,23531,14103,41236,86432,
%T 77644,64250,148534,243209,141005,384490,373985,29215,101281,543102,
%U 109281,154396,1122108,965630,1006716,1283207,152876,2147337,1419745,1545874,1381045,1108262,123879
%N a(n) = (p-1)! mod p^3, where p = prime(n).
%H Robert Israel, <a href="/A330526/b330526.txt">Table of n, a(n) for n = 1..10000</a>
%H Claire Levaillant, <a href="https://arxiv.org/abs/1912.06652">Wilson's theorem modulo p^2 derived from Faulhaber polynomials</a>, arXiv:1912.06652 [math.CO], 2019.
%H Zhi-Hong Sun, <a href="https://doi.org/10.1016/S0166-218X(00)00184-0">Congruences concerning Bernoulli numbers and Bernoulli polynomials</a>, Discrete Applied Math. 105 (2000) 193 - 223.
%F a(n)= A177771(n) mod A030078(n).
%p f:= proc(n) local p,p3,k,r;
%p p:= ithprime(n);
%p p3:= p^3;
%p r:= 1:
%p for k from 1 to p-1 do
%p r:= r*k mod p3
%p od;
%p r
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Dec 18 2019
%t Mod[(#-1)!,#^3]&/@Prime[Range[40]] (* _Harvey P. Dale_, Jan 09 2024 *)
%o (PARI) a(n) = my(p=prime(n)); (p-1)! % p^3;
%o (Magma) [Factorial(p-1)mod p^3: p in PrimesUpTo(170)]; // _Marius A. Burtea_, Dec 18 2019
%Y Cf. A030078, A112660, A177771.
%K nonn
%O 1,2
%A _Michel Marcus_, Dec 17 2019