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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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14
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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 asymptotically 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), which happens to be primes p s.t. (p, 2p+1) is a prime pair.
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.
Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional comments and references. [From Jonathan Sondow, Nov 18 2009]
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. [From 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
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REFERENCES
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R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 84-93
Philippe Flajolet and Ilan Vardi, Zeta function Expansions of Classical constants, Feb. 18, 1996
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).
J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,1001
C. K. Caldwell, The Prime Glossary, twin prime constant
H. Cohen, High-precision calculation of Hardy-Littlewood constants, 1998. - From N. J. A. Sloane, Jan 26 2013
S. R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1. [From Jonathan Sondow, Nov 18 2009]
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 constant T_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Niklasch, Twin primes constant
_Simon Plouffe_, The twin primes constant
_Simon Plouffe_, Plouffe's Inverter, The twin primes constant
Pascal Sebah (pascal_sebah(AT)ds-fr.com), Numbers, constants and computation (gives 5000 digits)
Eric Weisstein's World of Mathematics, Twin Primes Constant
Eric Weisstein's World of Mathematics, Twin Prime Conjecture
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
Eric Weisstein's World of Mathematics, Prime Constellation
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FORMULA
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prod(k>=2, (zeta(k)*(1-1/2^k))^(-sum(d/k, mu(d)*2^(k/d))/k)) - Benoit Cloitre, Aug 06 2003
Equals 1/A167864. [From Jonathan Sondow, Nov 18 2009]
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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 *)
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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))
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CROSSREFS
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Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271, A114907, A065418 (C_3).
Sequence in context: A002892 A055667 A155742 * A197013 A081825 A212708
Adjacent sequences: A005594 A005595 A005596 * A005598 A005599 A005600
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KEYWORD
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cons,nonn,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Vladeta Jovovic, Nov 08 2001
Fixed my PARI program, had -n Harry J. Smith, May 19 2009
Commented and edited by Daniel Forgues, Jul 28 2009, Aug 04 2009, Aug 12 2009
PARI code removed by D. S. McNeil, Dec 26 2010
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STATUS
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approved
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