A007619 -id:A007619 - cp's OEIS frontend

A163211 Swinging Wilson quotients (A163210) which are primes.

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

A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

Original entry on oeis.org

2, 12, 788, 7636, 874202, 10018884, 1445893544, 2954512034024, 38700329118256, 93229749133527532, 17540746936557672236, 243284404062970619608, 47694250379410432495952, 136236017676683906365850456, 404504597532158799519693872144, 5856120097210409121404621878992, 18102352585707069737371994385420772, 3894254646848417473467131712404310728

Offset: 3

Jonathan Sondow, Feb 22 2013

nonn

Morley (1894/95) proved 2^(2*p-2) == (-1)^((p-1)/2)*binomial(p-1,(p-1)/2) mod p^3 for all primes p > 3.

Morley quotients are even, since 2^(2*p-2) and binomial(p-1,(p-1)/2) are even and p^3 is odd.

			prime(3) = 5, so a(3) = (2^(2*5-2) - (-1)^((5-1)/2)*binomial(5-1,(5-1)/2))/5^3 = (2^8 - binomial(4,2))/5^3 = (256-6)/125 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
C. Aebi, G. Cairns, Morley’s other miracle, Math. Mag., 85 (2012), 205-211.
F. Morley, Note on the Congruence 2^4n == (-1)^n*(2n)!/(n!)^2 where 2n+1 is a prime, Annals of Mathematics, Vol. 9 (1894 - 1895), pp. 168-170.

Cf. A007619, A007663, A034602, A197630, A197633.

m[p_] := (2^(2*p-2) - (-1)^((p-1)/2)*Binomial[p-1, (p-1)/2])/p^3; Table[ m[ Prime[n]], {n, 3, 20}]

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)).

A239564 a(n) = (round(c^prime(n)) - 1)/prime(n), where c is the pentanacci constant (A103814).

Original entry on oeis.org

154, 504, 5758, 19912, 245714, 11251030, 40679232, 1967728552, 26525975822, 97753187576, 1335948880418, 68398141417510, 3547322151373882, 13260715720748120, 697034813138756392, 9825603574709578482, 36935066391752894480, 1970457739485406707872

Offset: 5

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Pentanacci Constant

Cf. A000040, A007619, A007663, A238693, A238697, A238698, A238700, A103814, A239502, A239544.

A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.

Original entry on oeis.org

1, 3, 101505, 259105757127, 1356566605613854774200240267, 1851197466245939272480116323530608949000567215

Offset: 1

Jaime Gómez, Sep 20 2017

nonn

Clement's criterion for twin primes is, for integers with n >= 2: n and n + 2 are both primes if and only if 4*((n-1)! + 1) + n == 0 (mod n*(n+2)). See the Clement and Ribenboim links. Like the criteron for primality using Theorem 81 of Hardy and Wright, p. 69, it "is of course quite useless as a practical test".
a(n) is an integer because of the necessary part of this twin prime criterion.
Thanks to Wolfdieter Lang for comments and helpful advice.

			a(2) = 3, because A001359(2) = 5 and C(5) = (4*(4! + 1) + 5)/(5*7) = 3.
a(2) = 3 because A014574(2) = 6 and delta(6) = (4*4! + 6 + 3)/35 = 3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, 1979.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260 (a proof of Clement's theorem).

Jaime Gómez, Table of n, a(n) for n = 1..23
P. A. Clement, Congruences for sets of primes, American Mathematical Monthly, vol. 56 (1949), pages 23-25.
L. Cong and Z. Li, On Wilson's Theorem and Polignac's Conjecture, arXiv:math/0408018 [math.NT], 2004.

Cf. A001359, A007619, A014574.

p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p);
a[n_] := (4*((p1[n] - 1)! + 1) + p1[n])/(p1[n]*(p1[n] + 2));
Array[a, 6] (* Jean-François Alcover, Nov 04 2017 *)

c(n) = (4*(n - 2)! + n + 3) / (n^2 - 1);
lista(nn) = forprime(p=2, nn, if (isprime(p+2), print1(c(p+1), ", "));); \\ Michel Marcus, Sep 21 2017

# Python version 2.7
import math
from sympy import *
list = []
n = 3
l = 1   # parameter that indicates the desired length of the list
x = 1
while x <= l:
       y = (4*factorial(n-2))+n+3
       z = n**2 - 1
       if y % z == 0:
              print (y/z)
              list.append(y/z)
       n+=1
       x+=1

a(n) = (4*((p1(n)-1)! + 1) + p1(n))/(p1(n)*(p1(n) + 2)) with p1(n) = A001359(n), for n >= 1. See the name.
From Wilson's theorem (see Hardy and Wright, Theorem 80, p. 68), a(n) = (4*kp1(n) + 1)/(p1(n) + 2) with p1(n) = A000359(n) and kp1(n) = A007619(p1(n)).
a(n) = delta(A014574(n)) with delta(n) = (4*(n-2)!+ n + 3)/(n^2 - 1).
delta(n) ~ ((4*(n-2)^(n - 2)* sqrt(2*Pi*(n - 2))) / (e^(n - 2)*(n^2 - 1)))+((n + 3) / (n^2 - 1)) for large n-values (using Stirling's approximation for n!).

Edited by Wolfdieter Lang, Oct 25 2017

A091330 a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!), where p is the n-th prime.

Original entry on oeis.org

0, 0, 4, 102, 329890, 36846276, 1230752346352, 336967037143578, 48869596859895986086, 10513391193507374500051862068, 8556543864909388988268015483870, 10053873697024357228864849950022572972972

Offset: 1

Russell Easterly, Mar 01 2004

Related to Wilson's Theorem. Let p be a prime number and write 1/p - (p-1)/p! = x/(p-1)!. Then x = (p-1)!/p - (p-1)*(p-1)!/p! = (p-1)!/p - (p-1)/p.
Also, a(n) = floor((p-1)!/p). [Bruno Berselli, May 31 2013]
If b(1)=1, and b(m) = ((m-1)^2 / m) *(b(m-1)+(m-3)/(m-1)) for m>1, then a(n) are the terms of b(m) for m prime. [Pedro Caceres, Dec 30 2018]

			Prime(4)=7 so a(4) = 6!/7 - 6*6!/7! = 102

Cf. A007619.

A091330[n_] := Block[{p = Prime[n]}, ((p - 1)!/p) - ((p - 1)*(p - 1)!/p!)] (* Robert G. Wilson v, Mar 02 2004 *)

More terms from Robert G. Wilson v and Ray Chandler, Mar 02 2004

A152413 Generalized Wilson primes of order 17; or primes p such that p^2 divides 16!(p-17)! + 1.

Original entry on oeis.org

61, 251, 479

Offset: 1

Alexander Adamchuk, Dec 03 2008

Wilson's theorem states that (p-1)! == -1 (mod p) for every prime p. Wilson primes are the primes p such that p^2 divides (p-1)! + 1. They are listed in A007540. Wilson's theorem can be expressed in general as (n-1)!(p-n)! == (-1)^n (mod p) for every prime p >= n. Generalized Wilson primes order n are the primes p such that p^2 divides (n-1)!(p-n)! - (-1)^n.

Alternatively, prime p=prime(k) is a generalized Wilson prime order n iff A002068(k) == A007619(k) == H(n-1) (mod p), where H(n-1) = A001008(n-1)/A002805(n-1) is (n-1)-st harmonic number. For this sequence (n=17), it reduces to A002068(k) == A007619(k) == 2436559/720720 (mod p).

Eric Weisstein's World of Mathematics, Wilson Prime

Cf. A007540, A007619, A079853, A124405, A128666.

Edited by Max Alekseyev, Jan 28 2012

A193447 a(n) = ((p - 2)! + p - 1)/(p*(p - 1)) where p is the n-th prime.

Original entry on oeis.org

3, 3299, 255877, 4807626353, 1040021719579, 100970241446066087, 13409937746820630739862069, 9507270961010432209186683871, 7757618593382991688938927430572972973, 12437732976339904486975781548721278876097561, 18522993694996570934756402022946152638511627907

Offset: 4

Alzhekeyev Ascar M, Jul 26 2011

nonn

Conjecture: for k >= 7, ((k - 2)! + k - 1)/(k*(k - 1)) is an integer iff k is prime.
Proof follows from Wilson's theorem. - Alois P. Heinz, Aug 07 2011
Note that a(1) = 1 is also an integer. - Jianing Song, Sep 17 2018

			a(4) = (5! + 6)/(7*6) = 126/42 = 3.
a(5) = (9! + 10)/(11*10) = 362890/110 = 3299.

Wikipedia, Wilson's theorem

Cf. A000040, A007619, A066161.

a(n) = my(p=prime(n)); ((p-2)!+p-1)/(p*(p-1)) \\ Jianing Song, Sep 17 2018

Name clarified by Jianing Song, Sep 17 2018

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.

A239565 (Round(c^prime(n)) - 1)/prime(n), where c is the hexanacci constant (A118427).

Original entry on oeis.org

6702, 23594, 301738, 14576792, 53653610, 2738173594, 38254296398, 143514673148, 2032676550562, 109797468019174, 6007838407290514, 22863415355711030, 1267938526864061370, 18523200405015238420, 70884650213591098558, 3989789924439684599434

Offset: 7

Vladimir Shevelev and Peter J. C. Moses, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Hexanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A118427, A239502, A239544, A239564.

cp's OEIS Frontend

A163211 Swinging Wilson quotients (A163210) which are primes.

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A222207 Morley quotients: (2^(2*p-2) - (-1)^((p-1)/2)*binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

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A225906 Indices of primes whose Wilson quotients are also prime.

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A239564 a(n) = (round(c^prime(n)) - 1)/prime(n), where c is the pentanacci constant (A103814).

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A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4*((n-1)! + 1) + n)/(n*(n+2)) for n >= 2.

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A091330 a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!), where p is the n-th prime.

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A152413 Generalized Wilson primes of order 17; or primes p such that p^2 divides 16!(p-17)! + 1.

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A193447 a(n) = ((p - 2)! + p - 1)/(p*(p - 1)) where p is the n-th prime.

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A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.