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A005597
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Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).
(Formerly M4056)
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37
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6, 6, 0, 1, 6, 1, 8, 1, 5, 8, 4, 6, 8, 6, 9, 5, 7, 3, 9, 2, 7, 8, 1, 2, 1, 1, 0, 0, 1, 4, 5, 5, 5, 7, 7, 8, 4, 3, 2, 6, 2, 3, 3, 6, 0, 2, 8, 4, 7, 3, 3, 4, 1, 3, 3, 1, 9, 4, 4, 8, 4, 2, 3, 3, 3, 5, 4, 0, 5, 6, 4, 2, 3, 0, 4, 4, 9, 5, 2, 7, 7, 1, 4, 3, 7, 6, 0, 0, 3, 1, 4, 1, 3, 8, 3, 9, 8, 6, 7, 9, 1, 1, 7, 7, 9
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OFFSET
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0,1
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COMMENTS
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C_2 = Product_{ p prime > 2} (p * (p-2) / (p-1)^2) is the 2-tuple case of the Hardy-Littlewood prime k-tuple constant (part of First H-L Conjecture): C_k = Product_{ p prime > k} (p^(k-1) * (p-k) / (p-1)^k).
Although C_2 is commonly called the twin prime constant, it is actually the prime 2-tuple constant (prime pair constant) which is relevant to prime pairs (p, p+2m), m >= 1.
The Hardy-Littlewood asymptotic conjecture for Pi_2m(n), the number of prime pairs (p, p+2m), m >= 1, with p <= n, claims that Pi_2m(n) ~ C_2(2m) * Li_2(n), where Li_2(n) = Integral_{2, n} (dx/log^2(x)) and C_2(2m) = 2 * C_2 * Product_{p prime > 2, p | m} (p-1)/(p-2), which gives: C_2(2) = 2 * C_2 as the prime pair (p, p+2) constant, C_2(4) = 2 * C_2 as the prime pair (p, p+4) constant, C_2(6) = 2* (2/1) * C_2 as the prime pair (p, p+6) constant, C_2(8) = 2 * C_2 as the prime pair (p, p+8) constant, C_2(10) = 2 * (4/3) * C_2 as the prime pair (p, p+10) constant, C_2(12) = 2 * (2/1) * C_2 as the prime pair (p, p+12) constant, C_2(14) = 2 * (6/5) * C_2 as the prime pair (p, p+14) constant, C_2(16) = 2 * C_2 as the prime pair (p, p+16) constant, ... and, for i >= 1, C_2(2^i) = 2 * C_2 as the prime pair (p, p+2^i) constant.
C_2 also occurs as part of other Hardy-Littlewood conjectures related to prime pairs, e.g., the Hardy-Littlewood conjecture concerning the distribution of the Sophie Germain primes (A156874) on primes p such that 2p+1 is also prime.
Another constant related to the twin primes is Viggo Brun's constant B (sometimes also called the twin primes Viggo Brun's constant B_2) A065421, where B_2 = Sum (1/p + 1/q) as (p,q) runs through the twin primes.
C_2 = Product_{prime p>2} (p-2)p/(p-1)^2 is an analog for primes of Wallis' product 2/Pi = Product_{n=1 to oo} (2n-1)(2n+1)/(2n)^2. - Jonathan Sondow, Nov 18 2009
One can compute a cubic variant, product_{primes >2} (1-1/(p-1)^3) = 0.855392... = (2/3) * 0.6601618...* 1.943596... by multiplying this constant with 2/3 and A082695. - R. J. Mathar, Apr 03 2011
Cohen (1998, p. 7) referred to this number as the "twin prime and Goldbach constant" and noted that, conjecturally, the number of twin prime pairs (p,p+2) with p <= X tends to 2*C_2*X/log(X)^2 as X tends to infinity. - Artur Jasinski, Feb 01 2021
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 84-93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Paul S. Bruckman, Problem H-576, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), p. 379; General IZE, Solution to Problem H-576 by the proposer, ibid., Vol. 40, No. 4 (2002), pp. 383-384.
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FORMULA
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Equals Product_{k>=2} (zeta(k)*(1-1/2^k))^(-Sum_{d|k} mu(d)*2^(k/d)/k). - Benoit Cloitre, Aug 06 2003
Equals Sum_{k>=1} mu(2*k-1)/phi(2*k-1)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010) (Bruckman, 2001). - Amiram Eldar, Jan 14 2022
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EXAMPLE
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0.6601618158468695739278121100145557784326233602847334133194484233354056423...
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MATHEMATICA
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s[n_] := (1/n)*N[ Sum[ MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[ (Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, 160}]; RealDigits[C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 15 2012, after PARI *)
digits = 105; f[n_] := -2*(2^n-1)/(n+1); C2 = Exp[NSum[f[n]*(PrimeZetaP[n+1] - 1/2^(n+1)), {n, 1, Infinity}, NSumTerms -> 5 digits, WorkingPrecision -> 5 digits]]; RealDigits[C2, 10, digits][[1]] (* Jean-François Alcover, Apr 16 2016, updated Apr 24 2018 *)
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PROG
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(PARI) \p1000; 175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))
(PARI) prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 12 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Commented and edited by Daniel Forgues, Jul 28 2009, Aug 04 2009, Aug 12 2009
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STATUS
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approved
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