A007619 Wilson quotients: ((p-1)! + 1)/p where p is the n-th prime. (Formerly M4023)

1, 1, 5, 103, 329891, 36846277, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

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.

Jonathan 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.

Jonathan 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

Cf. A005450, A005451, A007540 (Wilson primes), A050299, A163212, A225672, A225906.

Cf. A261779.

Cf. A157249, A157250, A292691 (twin prime analog quotient).

Sequence in context: A377474 A159523 A172116 * A163212 A163154 A354157

Adjacent sequences: A007616 A007617 A007618 * A007620 A007621 A007622

Keyword

nonn

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Extensions

Definition clarified by Jonathan Sondow, Aug 05 2011

Status

approved