A157249 -id:A157249 - cp's OEIS frontend

A157250 Wilson numbers: k such that the generalized Wilson quotient A157249(k) is divisible by k.

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158

Offset: 1

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

A prime p is a Wilson prime if p divides its Wilson quotient A007619. A number n is a Wilson number if n divides its generalized Wilson quotient A157249.
The sequence contains all Wilson numbers <= 5 x 10^8. Heuristics suggest that #(Wilson numbers < N) is about (6/pi^2) log N, for large N.
A Wilson number is prime if and only if it is a Wilson prime A007540. Only three are known: 5, 13, 563.
The first composite Wilson number 5971 was discovered by Kloss, the others by Agoh, Dilcher, and Skula. Every known composite Wilson number n has at least two odd prime factors, so e(n) = -1.
For additional references and links, see A007540.

			A157249(13) = (A001783(13) + e(13))/13 = ((13-1)! + 1)/13 = 479001601/13 = 36846277 == 0 mod 13, so 13 is a member. A001783(5971) + e(5971) = A001783(5971) - 1 == 0 mod 5971^2, so 5971 is a member. But A157249(8) = (A001783(8) + e(8))/8 = (3*5*7 - 1)/8 = 13 ==/== 0 mod 8, so 8 is not a member.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966, p. 65.

T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861.
K. E. Kloss, Some Number-Theoretic Calculations, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.
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. Wilson quotient A007619, Wilson prime A007540, generalized Wilson quotient A157249, n-phi-torial A001783, numbers having a primitive root A033948.

f[n_] := Times @@ Select[Range[n], CoprimeQ[n, #]&];
e[1|2|4] = 1; e[n_] := If[MatchQ[FactorInteger[n], {{?OddQ, }} | {{2, 1}, {, }}], 1, -1];
WilsonQ[n_] := IntegerQ[(f[n] + e[n])/n^2];
Reap[For[k = 1, k < 10^7, k++, If[WilsonQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 11 2018 *)

A157249(n) == 0 mod n.

A001783(n) + e(n) == 0 mod n^2, where e(n) = +1 or -1 according as n does or does not have a primitive root.

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

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

Original entry on oeis.org

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

A007540 Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

Original entry on oeis.org

5, 13, 563

Offset: 1

N. J. A. Sloane

Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p). Cf. Wilson quotients, A007619.
Sequence is believed to be infinite. Next term is known to be > 2*10^13 (cf. Costa et al., 2013).
Intersection of the Wilson numbers A157250 and the primes A000040. - Jonathan Sondow, Mar 04 2016
Conjecture: Odd primes p such that 1^(p-1) + 2^(p-1) + ... + (p-1)^(p-1) == p-1 (mod p^2). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
From Felix Fröhlich, Nov 16 2018: (Start)
Harry S. Vandiver apparently said about the Wilson primes "It is not known if there are infinitely many Wilson primes. This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me that it had definitely been settled, I think I would immediately drop dead again." (cf. Ribenboim, 2000, p. 217).
Let p be a Wilson prime and let i be the index of p in A000040. For n = 1, 2, 3, the values of i are 3, 6, 103. The primes among those values are Lerch primes, i.e., terms of A197632. Is this a property that necessarily follows if i is prime (cf. Sondow, 2011/2012, 2.5 Open Problems 5)? (End)
From Amiram Eldar, Jun 16 2021: (Start)
Named after the English mathematician John Wilson (1741-1793) after whom "Wilson's theorem" was also named.
The primes 5 and 13 appear in an exercise involving the Wilson congruence in Mathews (1892). [Edited by Felix Fröhlich, Jul 23 2021]
Beeger found that there are no other smaller terms up to 114 (1913) and up to 200 (1930).
a(3) = 563 was found by Goldberg (1953), who used the Bureau of Standards Eastern Automatic Computer (SEAC) to search all primes less than 10000. According to Goldberg, the third prime was discovered independently by Donald Wall six month later. (End)

N. G. W. H. Beeger, On the Congruence (p-1)! == -1 (mod p^2), Messenger of Mathematics, Vol. 49 (1920), pp. 177-178.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
Calvin C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 180.
Richard Crandall and Carl 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, 5th ed., Oxford Univ. Press, 1979, th. 80.
G. B. Mathews, Theory of Numbers Part I., Cambridge: Deighton, Bell and Co., London: George Bell and Sons, 1892, page 318.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer Science & Business Media, 2000, ISBN 0-387-98911-0.
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 pp. 234-235.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 73.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 163.

N. G. W. H. Beeger, Quelques remarques sur les congruences r^(p-1) == 1 (mod p^2) et (p- 1)! == -1 (mod p^2), The Messenger of Mathematics, Vol. 43 (1913), pp. 72-84.
Edgar Costa, Robert Gerbicz and David Harvey, A search for Wilson primes, Mathematics of Computation, Vol. 83, No. 290 (2014), pp. 3071-3091; arXiv preprint, arXiv:1209.3436 [math.NT], 2012.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.
Karl Goldberg, A Table of Wilson Quotients and the Third Wilson Prime, Journal of the London Mathematical Society, Vol. 28 (1953), pp. 252-256.
James Grime and Brady Haran, What do 5, 13 and 563 have in common?, YouTube video (2014).
Emma Lehmer, A Note on Wilson's Quotient, The American Mathematical Monthly, Vol. 44, No. 4 (1937), pp. 237-238.
Emma Lehmer, On the Congruence (p-1)! == -1 (mod p^2), The American Mathematical Monthly, Vol. 44, No. 7 (1937), p. 462.
Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson", Annals of Mathematics, Vol. 39, No. 2 (1938), pp. 350-360.
George Ballard Mathews, Theory of numbers, Part I, Cambridge, 1892, p. 318.
Tapio Rajala, Status of a search for Wilson primes
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: M. B. Nathanson, Combinatorial and Additive Number Theory, Springer, CANT 2011 and 2012. Also on arXiv, arXiv:1110.3113 [math.NT], 2011-2012.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Wilson Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Wilson prime.
Paul Zimmermann, Records for prime numbers.

Cf. A007619, A157249, A157250, A377266.

Select[Prime[Range[500]], Mod[(# - 1)!, #^2] == #^2 - 1 &] (* Harvey P. Dale, Mar 30 2012 *)

forprime(n=2, 10^9, if(Mod((n-1)!, n^2)==-1, print1(n, ", "))) \\ Felix Fröhlich, Apr 28 2014

is(n)=prod(k=2,n-1,k,Mod(1,n^2))==-1 \\ Charles R Greathouse IV, Aug 03 2014

from sympy import prime
A007540_list = []
for n in range(1,10**4):
    p, m = prime(n), 1
    p2 = p*p
    for i in range(2,p):
        m = (m*i) % p2
    if m == p2-1:
        A007540_list.append(p) # Chai Wah Wu, Dec 04 2014

A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 47, 53, 59, 61, 71, 107, 137, 149, 173, 227, 251, 277, 313, 347, 349, 359, 367, 373, 409, 419, 443, 463, 467, 479, 491, 499, 521, 523, 541, 563, 577, 599, 607, 613, 617, 631, 643, 647, 677, 683, 739, 751, 757, 809, 811, 821, 823, 827, 829

Offset: 1

Richard Peterson, Oct 22 2024

nonn

The generalization of Wilson's Theorem that x!*(p-1-x)! is either +1 or -1 mod p when p is prime is known. This is investigated here for cases of p such that there is an x with x!*(p-1-x)! congruent to either +1 or -1 mod p^2.

			0!*4! + 1 = 5^2 and 4+0 = 5-1, so 5 is in the sequence.
1!*9! - 1 = 11^2*2999 and 1+9 = 10-1, so 11 is in the sequence.
0!*12! + 1 = 13^2*2834329 and 0+12 = 13-1, so 13 is in the sequence.
10!*6! + 1 = 17^2*83*108923 and 10+6=17-1, so 17 is in the sequence.
19!*39!-1 is divisible by 59^2 and 19+39=59-1, so 59 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..4011

A007540 is a subsequence.

Cf. A157249, A157250.

filter:= proc(p) local m,n,A,v;
  A:= Array(0..p);
  A[0]:= 1:
  for n from 1 to p do A[n]:= n*A[n-1] mod p^2 od:
  for m from 0 to (p-1)/2 do
    v:= A[m] * A[p-1-m] mod p^2;
    if v = 1 or v = p^2-1 then return true fi;
  od;
  false
end proc:
select(filter, [seq(ithprime(i),i=1..150)]); # Robert Israel, Dec 30 2024

isok(p)={if(isprime(p), for(m=0, p\2, my(t=(m!*(p-1-m)!%p^2)); if(t==1||t==p^2-1, return(1)))); 0} \\ Andrew Howroyd, Oct 22 2024

a(14) onwards from Andrew Howroyd, Oct 22 2024

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A157250 Wilson numbers: k such that the generalized Wilson quotient A157249(k) is divisible by k.

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

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

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A007540 Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

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A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

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