This site is supported by donations to The OEIS Foundation.



Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)



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


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).


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.


a(n) = A157249(prime(n)). - Jonathan Sondow, Mar 04 2016


The 4th prime is 7, so a(4) = (6! + 1)/7 = 103.


Table[With[{p=Prime[n]}, ((p-1)!+1)/p], {n, 15}] (* Harvey P. Dale, Oct 16 2011 *)


(PARI) a(n)=my(p=prime(n)); ((p-1)!+1)/p \\ Charles R Greathouse IV, Apr 24 2015


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

Cf. A261779.

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

Sequence in context: A142418 A159523 A172116 * A163212 A163154 A165387

Adjacent sequences:  A007616 A007617 A007618 * A007620 A007621 A007622




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


Definition clarified by Jonathan Sondow, Aug 05 2011



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 06:54 EST 2017. Contains 294861 sequences.