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

1, 3, 101505, 259105757127, 1356566605613854774200240267, 1851197466245939272480116323530608949000567215

Offset

1,2

Comments

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.

References

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).

Links

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.

Formula

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!).

Example

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.

Mathematica

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 *)

Prog

(Python 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
(PARI) 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

Crossrefs

Cf. A001359, A007619, A014574.

Sequence in context: A119119 A164841 A171366 * A086785 A159577 A116536

Adjacent sequences: A292688 A292689 A292690 * A292692 A292693 A292694

Keyword

nonn

Author

Jaime Gómez, Sep 20 2017

Extensions

Edited by Wolfdieter Lang, Oct 25 2017

Status

approved