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%I A005597 M4056
%S A005597 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,
%T A005597 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,
%U A005597 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
%N A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime 
               >= 3 } (1-1/(p-1)^2).
%C A005597 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).
%C A005597 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.
%C A005597 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 A005597 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.
%C A005597 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 A005597 Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional 
               comments and references. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Nov 18 2009]
%C A005597 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 (jsondow(AT)alumni.princeton.edu), Nov 18 2009]
%D A005597 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005597 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 
               Springer, NY, 2001; see p. 11.
%D A005597 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, pp. 84-93
%D A005597 Philippe Flajolet and Ilan Vardi, Zeta function Expansions of Classical 
               constants, Feb. 18, 1996
%D A005597 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, ch. 22.20.
%D A005597 J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime 
               constant, Math. Comp., 15 (1961), 396-398.
%H A005597 Harry J. Smith, <a href="b005597.txt">Table of n, a(n) for n=0,...,1001</
               a>
%H A005597 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=TwinPrimeConstant">twin prime constant</a>
%H A005597 Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/
               publist.html">Zeta function expansions of some classical constants</
               a>
%H A005597 G. Niklasch, <a href="http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml">
               Some number theoretical constants: 1000-digit values</a>
%H A005597 G. Niklasch, <a href="http://www.gn-50uma.de/alula/essays/Moree/Moree-details.en.shtml#t05-twin">
               Twin primes constant</a>
%H A005597 Pascal Sebah (pascal_sebah(AT)ds-fr.com), <a href="http://numbers.computation.free.fr/
               Constants/Primes/twin.html">Numbers, constants and computation</a> 
               (gives 5000 digits)
%H A005597 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap90.html">The twin primes 
               constant</a>
%H A005597 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               twinprime.txt">The twin primes constant</a>
%H A005597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TwinPrimesConstant.html">Twin Primes Constant</a>
%H A005597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TwinPrimeConjecture.html">Twin Prime Conjecture</a>
%H A005597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               k-TupleConjecture.html">k-Tuple Conjecture</a>
%H A005597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeConstellation.html">Prime Constellation</a>
%H A005597 S. R. Finch, <a href="http://algo.inria.fr/csolve/erradd.pdf">Mathematical 
               Constants, Errata and Addenda</a>, Sec. 2.1. [From Jonathan Sondow 
               (jsondow(AT)alumni.princeton.edu), Nov 18 2009]
%F A005597 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
%F A005597 Equals 1/A167864. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Nov 18 2009]
%e A005597 0.6601618158468695739278121100145557784326233602847334133194484233354056423...
%o A005597 (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))
%o A005597 Contribution from Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 
               2009: (Start)
%o A005597 (PARI) { default(realprecision,1002); c2=\
%o A005597 0.66016181584686957392781211001455577843262336028473341331944842333\
%o A005597 5405642304495277143760031413839867911779005226693304002965847755123\
%o A005597 3662277471657132139869687410976206302141537354348531315960978036699\
%o A005597 3213525529976719930247459059310108297829155383446929750520591665713\
%o A005597 3653611991532464281301172462306379341060056466676584434063501649322\
%o A005597 7235289680109349664756004788123579627894598424336557493755818548141\
%o A005597 7362867809870596949870384124336338658931196907915004057371781437108\
%o A005597 1810615401233104810577794415613125444598860988997585328984038108718\
%o A005597 0355252617198871121363828087823497223742240971426974417644552252655\
%o A005597 4899482977179097778404375789195659064999456706290782860882839599039\
%o A005597 4287082529070521554595671723599449769037800675978761690802426600295\
%o A005597 7110920996337082725592846721298580011486979418554018246398874939417\
%o A005597 1182852838236599705032872570808798066220106863047430520199239428201\
%o A005597 4311102297265141514194258422242375342296879836738796224286600285358\
%o A005597 098482833679152235700192585875285961205994728621007171131607980572; x=10*c2; 
               for (n=0, 1001, d=floor(x); x=(x-d)*10; write("b005597.txt", n, " 
               ", d)); } (End)
%Y A005597 Cf. A065645 (continued fraction), A065646 (denominators of convergents 
               to twin prime constant), A065647 (numerators of convergents to twin 
               prime constant), A062270, A062271.
%Y A005597 Cf. A114907 Decimal expansion of twice the twin primes constant defined 
               in A005597.
%Y A005597 Cf. A065421 Decimal expansion of Viggo Brun's constant B, also known 
               as the twin primes Brun's constant B_2.
%Y A005597 Sequence in context: A002892 A055667 A155742 this_sequence A081825 A028969 
               A029681
%Y A005597 Adjacent sequences: A005594 A005595 A005596 this_sequence A005598 A005599 
               A005600
%K A005597 cons,nonn,nice
%O A005597 0,1
%A A005597 N. J. A. Sloane (njas(AT)research.att.com).
%E A005597 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 08 2001
%E A005597 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009
%E A005597 Commented and edited by Daniel Forgues (squid(AT)zensearch.com), Jul 
               28 2009, Aug 04 2009, Aug 12 2009

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