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A007619 Wilson quotients: ((p-1)!+1)/p where p is the n-th prime.
(Formerly M4023)
35
1, 1, 5, 103, 329891, 36846277, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.

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; http://www.scirp.org/journal/apm; http://dx.doi.org/10.4236/apm.2016.66028

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

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

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

Cf. A261779.

Cf. A157249, A157250.

Sequence in context: A142418 A159523 A172116 * A163212 A163154 A165387

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

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Last modified June 22 12:24 EDT 2017. Contains 288613 sequences.