A163212 -id:A163212 - 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

A163210 Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623

Offset: 1

Peter Luschny, Jul 24 2009

nonn

			The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..470
M. E. Bassett, S. Majid, Finite noncommutative geometries related to F_p[x], arXiv:1603.00426 [math.QA], 2016.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163213, A002068, A163212, A163209, A007619, A007540.

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WQ := proc(f,r,n) map(p->(f(p-1)+r(p))/p,select(isprime,[$1..n])) end:
A163210 := n -> WQ(swing,p->(-1)^iquo(p+2,2),n);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 28 2013 *)
a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
Join[{1, 1}, a[Prime[Range[3,20]]]] (* Peter Luschny, May 13 2017 *)

a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017

A122696 Primes of the form ((k-1)! + 1)/k.

Original entry on oeis.org

2, 5, 103, 329891, 10513391193507374500051862069

Offset: 1

Alexander Adamchuk, Sep 22 2006

nonn

A163212, Wilson quotients (A007619: ((p-1)!+1)/p) which are primes, is a subsequence. Corresponding numbers n such that ((n-1)! + 1)/n is prime are listed in A050299 = {1, 5, 7, 11, 29, 773, 1321, 2621, ...}. a(6) has 1893 digits. a(7) has 3545 digits. a(8) has 7817 digits.

Except for a(1) = 2, same as A163212. - Jonathan Sondow, May 20 2013

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.

A050299 is the main entry for this sequence.

Cf. A007619, A163212.

Select[Table[((k-1)!+1)/k,{k,30}],PrimeQ] (* James C. McMahon, Nov 09 2024 *)

is(n)=isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015 \\ This program actually produces A050299 - Michel Marcus, Aug 02 2016

for(n=1, 1e2, if(((n-1)!+1)%n==0 && isprime(k=((n-1)!+1)/n), print1(k, ", "))) \\ Altug Alkan, Nov 22 2015

a(n) = A163212(n-1) = ((A050299(n)-1)! + 1)/A050299(n). - Jonathan Sondow, May 19 2013

The next term is too large to include.

a(4) and first comment corrected by Gionata Neri, Aug 02 2016

A163211 Swinging Wilson quotients (A163210) which are primes.

Original entry on oeis.org

3, 23, 71, 757, 30671, 1383331, 245273927, 3362110459, 107752663194272623, 5117886516250502670227, 34633371587745726679416744736000996167729085703, 114326045625240879227044995173712991937709388241980425799

Offset: 1

Peter Luschny, Jul 24 2009

nonn

a(14)-a(18) certified prime by Primo 4.2.0. a(17) = A163210(569) = P1239, a(18) = A163210(787) = P1812. - Charles R Greathouse IV, Dec 11 2016

			The quotient (252+1)/11 = 23 is a swinging Wilson quotient and a prime, so 23 is a member.

G. C. Greubel, Table of n, a(n) for n = 1..16
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163210, A163213, A163212, A163209, A007619.

A163211 := n -> select(isprime,A163210(n));

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Select[PrimeQ][Table[a[n], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2016 *)

sf(n)=n!/(n\2)!^2
forprime(p=2,1e3, t=sf(p-1)\/p; if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Dec 11 2016

cp's OEIS Frontend

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

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A163210 Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).

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A122696 Primes of the form ((k-1)! + 1)/k.

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A163211 Swinging Wilson quotients (A163210) which are primes.

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