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A007619
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Wilson quotients: ((p-1)!+1)/p where p is the n-th prime.
(Formerly M4023)
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20
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1, 1, 5, 103, 329891, 36846277, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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).
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REFERENCES
| R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
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).
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.
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LINKS
| 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
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EXAMPLE
| The 4th prime is 7, so a(4) = (6!+1)/7 = 103.
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MATHEMATICA
| Table[With[{p=Prime[n]}, ((p-1)!+1)/p], {n, 15}] (* From Harvey P. Dale, Oct 16 2011 *)
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CROSSREFS
| Cf. A005450, A005451, A007540 (Wilson primes).
Sequence in context: A052138 A142418 A159523 * A163212 A163154 A165387
Adjacent sequences: A007616 A007617 A007618 * A007620 A007621 A007622
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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EXTENSIONS
| Definition clarified by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 05 2011
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