reviewed

approved

proposed

reviewed

editing

proposed

a(n) = Sum[ _{k=1..n} ((n-k)!*(k-1)! - (-1)^k, {k,1,n} ]).

According to the Generalized Wilson-Lagrange Theorem , a prime p divides (p-k)!*(k-1)! - (-1)^k for all integer integers k > 0. p divides a(p) for prime p. Quotients a(p)/p are listed in A134296(n) = {1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, ...}. p^2 divides a(p) for prime p = {7, 71}.

a(n) = Sum[ _{k=1..n} ((n-k)!*(k-1)! - (-1)^k, {k,1,n} ]).

Cf. A007540, A007619 = (Wilson quotients: ((p-1)!+1)/p. Cf. A134296 = Quotients a(p)/p.

Cf. A134296 (quotients a(p)/p).

approved

editing

_Alexander Adamchuk (alex(AT)kolmogorov.com), _, Oct 17 2007

Sum[ (n-k)!*(k-1)! - (-1)^k, {k,1,n} ].

2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, 89441280, 1060369921, 13649610240, 189550368001, 2824077312000, 44927447040001, 760034451456000, 13622700994560001, 257872110354432000, 5140559166898176001

1,1

According to the Generalized Wilson-Lagrange Theorem a prime p divides (p-k)!*(k-1)! - (-1)^k for all integer k>0. p divides a(p) for prime p. Quotients a(p)/p are listed in A134296(n) = {1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, ...}. p^2 divides a(p) for prime p = {7, 71}.

a(n) = Sum[ (n-k)!*(k-1)! - (-1)^k, {k,1,n} ].

Table[ Sum[ (n-k)!*(k-1)! - (-1)^k, {k, 1, n} ], {n, 1, 30} ]

Cf. A007540, A007619 = Wilson quotients:((p-1)!+1)/p. Cf. A134296 = Quotients a(p)/p.

nonn,new

Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 17 2007

approved