A005450 -id:A005450 - 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

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60

Offset: 1

N. J. A. Sloane

Denominator of (1 + Gamma(n))/n.

Möbius transform of A380441(n). - Wesley Ivan Hurt, Jun 21 2025

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Achilleas Sinefakopoulos, Problem 10578 (Submitted solution.)
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005171, A005450 (numerators).

Cf. A088140, A089026, A135684, A181569.

[IsPrime(n) select 1 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

[Denominator((1 + Factorial(n-1))/n): n in [1..70]]; // G. C. Greubel, Nov 22 2022

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 70}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)
Table[Denominator[(1 + Gamma[n])/n], {n,2,70}] (* G. C. Greubel, Nov 22 2022 *)

def A005451(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A005451(n) for n in (1..71)]   # Peter Luschny, Oct 16 2013

[denominator((1+gamma(n))/n) for n in range(1,71)] # G. C. Greubel, Nov 22 2022

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; a(n) = denominator(b(n)).
a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n^c(n), where c = A005171.
a(n) = Sum_{d|n} A380441(d) * mu(n/d). (End)

Name edited and a(1)=1 prepended by G. C. Greubel, Nov 22 2022. Name further edited by N. J. A. Sloane, Nov 22 2022

cp's OEIS Frontend

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

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