0,1

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.

Harry J. Smith, Table of n, a(n) for n=0,...,1001

C. K. Caldwell, The Prime Glossary, twin prime constant

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

G. Niklasch, Some number theoretical constants: 1000-digit values

G. Niklasch, Twin primes constant

Pascal Sebah (pascal_sebah(AT)ds-fr.com), Numbers, constants and computation (gives 5000 digits)

S. Plouffe, The twin primes constant

S. Plouffe, Plouffe's Inverter, The twin primes constant

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

prod(k>=2, (zeta(k)*(1-1/2^k))^(-sum(d/k, mu(d)*2^(k/d))/k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2003

0.6601618158468695739278121100145557784326233602847334133194484233354056423...

(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))

Contribution from Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009: (Start)

(PARI) { default(realprecision, 1002); c2=\

0.66016181584686957392781211001455577843262336028473341331944842333\

5405642304495277143760031413839867911779005226693304002965847755123\

3662277471657132139869687410976206302141537354348531315960978036699\

3213525529976719930247459059310108297829155383446929750520591665713\

3653611991532464281301172462306379341060056466676584434063501649322\

7235289680109349664756004788123579627894598424336557493755818548141\

7362867809870596949870384124336338658931196907915004057371781437108\

1810615401233104810577794415613125444598860988997585328984038108718\

0355252617198871121363828087823497223742240971426974417644552252655\

4899482977179097778404375789195659064999456706290782860882839599039\

4287082529070521554595671723599449769037800675978761690802426600295\

7110920996337082725592846721298580011486979418554018246398874939417\

1182852838236599705032872570808798066220106863047430520199239428201\

4311102297265141514194258422242375342296879836738796224286600285358\

098482833679152235700192585875285961205994728621007171131607980572; x=10*c2; for (n=0, 1001, d=floor(x); x=(x-d)*10; write("b005597.txt", n, " ", d)); } (End)

Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271.

Cf. A114907 Decimal expansion of twice the twin primes constant defined in A005597.

Cf. A065421 Decimal expansion of Viggo Brun's constant B, also known as the twin primes Brun's constant B_2.

Adjacent sequences: A005594 A005595 A005596 this_sequence A005598 A005599 A005600

Sequence in context: A002892 A055667 A155742 this_sequence A081825 A028969 A029681

cons,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 08 2001

Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009

Commented and edited by Daniel Forgues (squid(AT)zensearch.com), Jul 28 2009, Aug 04 2009, Aug 12 2009