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A330526 a(n) = (p-1)! mod p^3, where p = prime(n).

%I A330526 #18 Jan 09 2024 18:20:41
%S A330526 1,2,24,34,494,675,4419,4008,4944,13136,21730,23531,14103,41236,86432,
%T A330526 77644,64250,148534,243209,141005,384490,373985,29215,101281,543102,
%U A330526 109281,154396,1122108,965630,1006716,1283207,152876,2147337,1419745,1545874,1381045,1108262,123879
%N A330526 a(n) = (p-1)! mod p^3, where p = prime(n).
%H A330526 Robert Israel, <a href="/A330526/b330526.txt">Table of n, a(n) for n = 1..10000</a>
%H A330526 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 A330526 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 A330526 a(n)= A177771(n) mod A030078(n).
%p A330526 f:= proc(n) local p,p3,k,r;
%p A330526     p:= ithprime(n);
%p A330526     p3:= p^3;
%p A330526     r:= 1:
%p A330526     for k from 1 to p-1 do
%p A330526       r:= r*k mod p3
%p A330526     od;
%p A330526     r
%p A330526 end proc:
%p A330526 map(f, [$1..100]); # _Robert Israel_, Dec 18 2019
%t A330526 Mod[(#-1)!,#^3]&/@Prime[Range[40]] (* _Harvey P. Dale_, Jan 09 2024 *)
%o A330526 (PARI) a(n) = my(p=prime(n)); (p-1)! % p^3;
%o A330526 (Magma) [Factorial(p-1)mod p^3: p in PrimesUpTo(170)]; // _Marius A. Burtea_, Dec 18 2019
%Y A330526 Cf. A030078, A112660, A177771.
%K A330526 nonn
%O A330526 1,2
%A A330526 _Michel Marcus_, Dec 17 2019