|
%I M4023 #68 Oct 27 2017 22:10:52
%S 1,1,5,103,329891,36846277,1230752346353,336967037143579,
%T 48869596859895986087,10513391193507374500051862069,
%U 8556543864909388988268015483871,10053873697024357228864849950022572972973,19900372762143847179161250477954046201756097561,32674560877973951128910293168477013254334511627907
%N Wilson quotients: ((p-1)! + 1)/p where p is the n-th prime.
%C Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p).
%C 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 Subsequence of the generalized Wilson quotients A157249. - _Jonathan Sondow_, Mar 04 2016
%C 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 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford Science Publications, Clarendon Press, Oxford, 2003.
%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert G. Wilson v, <a href="/A007619/b007619.txt">Table of n, a(n) for n = 1..100</a>
%H 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 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 J. 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 J. 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 H. S. Wilf, <a href="http://www.jstor.org/stable/2974795">Problem 10578</a>, Amer. Math. Monthly, 104 (1997), 270.
%F a(n) = A157249(prime(n)). - _Jonathan Sondow_, Mar 04 2016
%e The 4th prime is 7, so a(4) = (6! + 1)/7 = 103.
%t Table[With[{p=Prime[n]},((p-1)!+1)/p],{n,15}] (* _Harvey P. Dale_, Oct 16 2011 *)
%o (PARI) a(n)=my(p=prime(n)); ((p-1)!+1)/p \\ _Charles R Greathouse IV_, Apr 24 2015
%Y Cf. A005450, A005451, A007540 (Wilson primes), A050299, A163212, A225672, A225906.
%Y Cf. A261779.
%Y Cf. A157249, A157250, A292691 (twin prime analog quotient).
%K nonn
%O 1,3
%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_
%E Definition clarified by _Jonathan Sondow_, Aug 05 2011
|
