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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. 3
1, 3, 101505, 259105757127, 1356566605613854774200240267, 1851197466245939272480116323530608949000567215 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A119119 A164841 A171366 * A086785 A159577 A116536
KEYWORD
nonn
AUTHOR
Jaime Gómez, Sep 20 2017
EXTENSIONS
Edited by Wolfdieter Lang, Oct 25 2017
STATUS
approved

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Last modified March 28 14:20 EDT 2024. Contains 371254 sequences. (Running on oeis4.)