This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A134296 #6 Jun 07 2021 04:44:29
%S A134296 1,2,13,259,750371,81566917,2642791002353,716984262871579,
%T A134296 102688143363690674087,21841112114495269555043222069,
%U A134296 17727866746681961093761724283871
%N A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).
%C A134296 A134295(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k) = {2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, ...}. According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. a(n) = A134295(p)/p for p = prime(n). a(n) is prime for n = {2, 3, 7, 9, 37, ...}. Corresponding prime terms in a(n) are {2, 13, 2642791002353, 102688143363690674087, ...}.
%F A134296 a(n) = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k) where p = prime(n).
%t A134296 Table[ (Sum[ (Prime[n]-k)!*(k-1)! - (-1)^k, {k,1,Prime[n]} ]) / Prime[n], {n,1,20} ]
%Y A134296 Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).
%Y A134296 Cf. A134295 (Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k)).
%K A134296 nonn
%O A134296 1,2
%A A134296 _Alexander Adamchuk_, Oct 17 2007