A050299 -id:A050299 - cp's OEIS frontend

A092029 Duplicate of A050299.

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

Offset: 1

dead

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

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

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

A084750 Numbers k such that k! - p is a prime, where p is the smallest prime > k.

Original entry on oeis.org

4, 5, 10, 11, 12, 14, 29, 53, 81, 90, 116, 236, 323, 346, 1172, 2957, 8400, 14906

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 16 2003

Numbers k such that k! - NextPrime(k) is prime.

If k != 3, there does not exist a prime p and a number k such that k! - NextPrime(k) < p < k! - 1. - Farideh Firoozbakht, Feb 26 2004

			10 is in the sequence because 10! = 3628800, NextPrime(10) = 11 and 3628800 - 11 = 3628789 is prime.

Cf. A084749, A050299, A002982, A064401, A084751.

Do[If[PrimeQ[k!-NextPrime[k]], Print[k]], {k, 0, 1425}] (* Farideh Firoozbakht, Feb 26 2004 *)

More terms from Farideh Firoozbakht, Feb 26 2004
a(16) from Ryan Propper, Jul 09 2005
Edited by N. J. A. Sloane at the suggestion of Ryan Propper, Jan 26 2008
a(17) from Michael S. Branicky, Jun 21 2023
a(18) from Michael S. Branicky, Apr 28 2025

A225906 Indices of primes whose Wilson quotients are also prime.

Original entry on oeis.org

3, 4, 5, 10, 137, 216, 381

Offset: 1

Jonathan Sondow, May 20 2013

nonn

Is it a coincidence that the terms are alternately odd and even? Is it also a coincidence that the odd terms are all primes (= A225672)?

			The Wilson quotient of 7 is ((7-1)!+1)/7 = 103, which is prime, and 7 is the 4th prime, so 4 is a member.

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
J. 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. A000720, A007619, A050299, A122696, A225672.

a(n) = A000720(A050299(n+1)).

A225672 Primes p such that the Wilson quotient of the p-th prime is also prime.

Original entry on oeis.org

3, 5, 137, 381

Offset: 1

Jonathan Sondow, May 20 2013

			The 5th prime is 11 and the Wilson quotient of 11 is ((11-1)!+1)/11 = 329891, which is prime, so 5 is a term.

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
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. A007619, A050299, A122696.

A317507 Numbers k whose generalized Wilson quotient A157249(k) is prime.

Original entry on oeis.org

1, 5, 7, 8, 10, 11, 29, 62, 486, 614, 773, 1321, 1906, 2621

Offset: 1

Amiram Eldar, Sep 29 2018

The corresponding primes are 2, 5, 103, 13, 19, 329891, ...
Supersequence of A050299 (except for 1, the prime terms of this sequence).
No more terms below 10^4.

Cf. A007619, A050299, A157249.

p[n_] := Times @@ Select[Range[n], CoprimeQ[n, #] &]; e[1 | 2 | 4] = 1; e[n_] := (fi = FactorInteger[n]; If[MatchQ[fi, {{(p_)?OddQ, }} | {{2, 1}, {, }}], 1, -1]); a[n] := (p[n] + e[n])/n; n = 1; s={}; Do[If[PrimeQ[a[n]], AppendTo[s,n]], {n, 1, 1000}]; s (* after Jean-François Alcover at A157249 *)

phito(n) = prod(k=2, n-1, k^(gcd(k, n)==1)); \\ A001783
is(n) = if(n%2, isprimepower(n) || n==1, n==2 || n==4 || (isprimepower(n/2, &n) && n>2)); \\ A033948
e(n) = if (is(n), 1, -1);
gw(n) = (phito(n)+e(n))/n;
isok(n) = isprime(gw(n)); \\ Michel Marcus, Oct 28 2018

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A092029 Duplicate of A050299.

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

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A163212 Wilson quotients (A007619) which are primes.

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

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A084750 Numbers k such that k! - p is a prime, where p is the smallest prime > k.

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A225906 Indices of primes whose Wilson quotients are also prime.

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A225672 Primes p such that the Wilson quotient of the p-th prime is also prime.

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A317507 Numbers k whose generalized Wilson quotient A157249(k) is prime.

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