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

%I A007619 M4023 #77 Apr 25 2025 04:27:21
%S A007619 1,1,5,103,329891,36846277,1230752346353,336967037143579,
%T A007619 48869596859895986087,10513391193507374500051862069,
%U A007619 8556543864909388988268015483871,10053873697024357228864849950022572972973,19900372762143847179161250477954046201756097561,32674560877973951128910293168477013254334511627907
%N A007619 Wilson quotients: ((p-1)! + 1)/p where p is the n-th prime.
%C A007619 Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p).
%C A007619 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).
%C A007619 Subsequence of the generalized Wilson quotients A157249. - _Jonathan Sondow_, Mar 04 2016
%C A007619 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
%D A007619 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
%D A007619 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford Science Publications, Clarendon Press, Oxford, 2003.
%D A007619 Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
%D A007619 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 234.
%D A007619 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007619 Robert G. Wilson v, <a href="/A007619/b007619.txt">Table of n, a(n) for n = 1..100</a>
%H A007619 Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, <a href="http://dx.doi.org/10.4236/apm.2016.66028">Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties</a>, Advances in Pure Mathematics, 2016, 6, 409-419.
%H A007619 Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H A007619 Jonathan Sondow, <a href="http://arxiv.org/abs/1110.3113">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
%H A007619 Jonathan Sondow, <a href="http://dx.doi.org/10.1007/978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
%H A007619 H. S. Wilf, <a href="http://www.jstor.org/stable/2974795">Problem 10578</a>, Amer. Math. Monthly, 104 (1997), 270.
%F A007619 a(n) = A157249(prime(n)). - _Jonathan Sondow_, Mar 04 2016
%e A007619 The 4th prime is 7, so a(4) = (6! + 1)/7 = 103.
%t A007619 Table[With[{p=Prime[n]},((p-1)!+1)/p],{n,15}] (* _Harvey P. Dale_, Oct 16 2011 *)
%o A007619 (PARI) a(n)=my(p=prime(n)); ((p-1)!+1)/p \\ _Charles R Greathouse IV_, Apr 24 2015
%Y A007619 Cf. A005450, A005451, A007540 (Wilson primes), A050299, A163212, A225672, A225906.
%Y A007619 Cf. A261779.
%Y A007619 Cf. A157249, A157250, A292691 (twin prime analog quotient).
%K A007619 nonn
%O A007619 1,3
%A A007619 _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_
%E A007619 Definition clarified by _Jonathan Sondow_, Aug 05 2011