A225906 -id:A225906 - cp's OEIS frontend

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

1, 1, 5, 103, 329891, 36846277, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

Offset: 1

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

nonn

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

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

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.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 234.
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.
Jonathan 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.
Jonathan 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.

Cf. A005450, A005451, A007540 (Wilson primes), A050299, A163212, A225672, A225906.
Cf. A261779.
Cf. A157249, A157250, A292691 (twin prime analog quotient).

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

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

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

Definition clarified by Jonathan Sondow, Aug 05 2011

A050299 Numbers k such that ((k-1)! + 1)/k is prime.

Original entry on oeis.org

1, 5, 7, 11, 29, 773, 1321, 2621

Offset: 1

N. J. A. Sloane, Apr 09 2003

Except for the first term, all terms are primes because for n > 1, n divides (n-1)! + 1 iff n is prime. - Farideh Firoozbakht, Mar 19 2004

a(9) >= 30941.

			7 is in the sequence because (6!+1)/7=103 is prime.

Mike Oakes, posting to Number Theory List, Aug 20 2003
Jonathan 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.
Jonathan 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.
Javier Soria, posting to Number Theory List, Apr 08 2003

v={1};Do[If[PrimeQ[((Prime[n]-1)!+1)/Prime[n]], v=Append[v, Prime[n]];Print[v]], {n, 845}]
Select[Range[2630],PrimeQ[((#-1)!+1)/#]&] (* Harvey P. Dale, Aug 18 2024 *)

is(n)=((n-1)!+1)%n==0 && isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015

((a(n)-1)! + 1)/a(n) = A122696(n) = A007619(A000720(A050299(n))) for n > 1. - Jonathan Sondow, Aug 07 2011

a(n) = prime(A225906(n-1)) for n > 1. - Jonathan Sondow, May 20 2013

a(7)-a(8) from Mike Oakes, Aug 20 2003

A163212 Wilson quotients (A007619) which are primes.

Original entry on oeis.org

5, 103, 329891, 10513391193507374500051862069

Offset: 1

Peter Luschny, Jul 24 2009

a(5) = A007619(137), a(6) = A007619(216), a(7) = A007619(381).

Same as A122696 without its initial term 2. - Jonathan Sondow, May 19 2013

			The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Jonathan 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.
Jonathan 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.

Cf. A050299, A163211, A007619, A122696, A163210, A163213, A163209, A225906.

# WQ defined in A163210.
A163212 := n -> select(isprime,WQ(factorial,p->1,n)):

Select[Table[p = Prime[n]; ((p-1)!+1)/p, {n, 1, 15}], PrimeQ] (* Jean-François Alcover, Jun 28 2013 *)

forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ Felix Fröhlich, Aug 03 2014

a(n) = A122696(n+1) = A007619(A225906(n)) = ((A050299(n+1)-1)!+1)/A050299(n+1). - Jonathan Sondow, May 19 2013

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

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A050299 Numbers k such that ((k-1)! + 1)/k is prime.

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A163212 Wilson quotients (A007619) which are primes.

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