OFFSET
1,3
COMMENTS
Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p).
Define b(n) = ((n-1)*(n^2 - 3*n + 1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; sequence gives b(primes).
Subsequence of the generalized Wilson quotients A157249. - Jonathan Sondow, Mar 04 2016
a(n) is an integer because of to Wilson's theorem (Theorem 80, p. 68, the if part of Theorem 81, p. 69, given in Hardy and Wright). See the first comment. `This theorem is of course quite useless as a practical test for the primality of a given number n' ( op. cit., p. 69). - Wolfdieter Lang, Oct 26 2017
REFERENCES
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford Science Publications, Clarendon Press, Oxford, 2003.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..100
Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.
FORMULA
a(n) = A157249(prime(n)). - Jonathan Sondow, Mar 04 2016
EXAMPLE
The 4th prime is 7, so a(4) = (6! + 1)/7 = 103.
MATHEMATICA
Table[With[{p=Prime[n]}, ((p-1)!+1)/p], {n, 15}] (* Harvey P. Dale, Oct 16 2011 *)
PROG
(PARI) a(n)=my(p=prime(n)); ((p-1)!+1)/p \\ Charles R Greathouse IV, Apr 24 2015
CROSSREFS
KEYWORD
nonn
EXTENSIONS
Definition clarified by Jonathan Sondow, Aug 05 2011
STATUS
approved