A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).

1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, 21841112114495269555043222069, 17727866746681961093761724283871

Offset

1,2

Comments

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, ...}.

Links

Table of n, a(n) for n=1..11.

Formula

a(n) = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k) where p = prime(n).

Mathematica

Table[ (Sum[ (Prime[n]-k)!*(k-1)! - (-1)^k, {k, 1, Prime[n]} ]) / Prime[n], {n, 1, 20} ]

Crossrefs

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

Cf. A134295 (Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k)).

Sequence in context: A276744 A042677 A338424 * A086510 A356493 A334089

Adjacent sequences: A134293 A134294 A134295 * A134297 A134298 A134299

Keyword

nonn

Author

Alexander Adamchuk, Oct 17 2007

Status

approved