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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. 2
 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 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)*(n² - 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 = 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

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Last modified March 18 13:47 EDT 2019. Contains 321289 sequences. (Running on oeis4.)