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 A065421 Decimal expansion of Viggo Brun's constant B, also known as the twin primes constant B_2: Sum (1/p + 1/q) as (p,q) runs through the twin primes. 25
 1, 9, 0, 2, 1, 6, 0, 5, 8 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The calculation of Brun's constant is "based on heuristic considerations about the distribution of twin primes" (Ribenboim, 1989). Another constant related to the twin primes is the twin primes constant C_2 (sometimes also denoted PI_2) A005597 defined in connection with the Hardy-Littlewood conjecture concerning the distribution pi_2(x) of the twin primes. Comment from Hans Havermann, Aug 06 2018: "I don't think the last three (or possibly even four) OEIS terms [he is referring to the sequence at that date - it has changed since then] are necessarily warranted. P. Sebah (see link below) (http://numbers.computation.free.fr/Constants/Primes/twin.html) gives 1.902160583104... as the value for primes to 10^16 followed by a suggestion that the (final) value 'should be around 1.902160583...'" - added by N. J. A. Sloane, Aug 06 2018 REFERENCES R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 14. S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 133-135. P. Ribenboim, The Book of Prime Number Records, 2nd. ed., Springer-Verlag, New York, 1989, p. 201. LINKS V. Brun, La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie, Bull Sci. Math. 43 (1919), 100-104 and 124-128. C. K. Caldwell, The Prime Glossary, Brun's constant Steven R. Finch, Brun's Constant [Broken link] Steven R. Finch, Brun's Constant [From the Wayback machine] Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204. Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only] Thomas R. Nicely, Prime Constellations Research Project P. Sebah, Numbers, constants and computation D. Shanks and J. W. Wrench, Brun's constant, Math. Comp. 28 (1974) 293-299; 28 (1974) 1183; Math. Rev. 50 #4510. H. Tronnolone, A tale of two primes, COLAUMS Space, #3, 2013. Wikipedia, Brun's constant FORMULA Equals Sum_{n>=1} 1/A077800(n). From Dimitris Valianatos, Dec 21 2013: (Start) (1/5) + Sum_{n>=1, excluding twin primes 3,5,7,11,13,...} mu(n)/n = (1/5) + 1 - 1/2 + 1/6 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 - 1/23 + 1/26 - 1/30 + 1/33 + 1/34 + 1/35 - 1/37 + 1/38 + 1/39 - 1/42 ... = 1.902160583... (End) EXAMPLE (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... = 1.902160583209 +- 0.000000000781 [Nicely] CROSSREFS Cf. A005597 (twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2)). Cf. A077800 (twin primes). Sequence in context: A221507 A089481 A188593 * A198556 A261169 A093767 Adjacent sequences: A065418 A065419 A065420 * A065422 A065423 A065424 KEYWORD hard,more,nonn,cons,nice AUTHOR Robert G. Wilson v, Sep 08 2000 EXTENSIONS Corrected by N. J. A. Sloane, Nov 16 2001 More terms computed by Pascal Sebah (pascal_sebah(AT)ds-fr.com), Jul 15 2001 Further terms computed by Pascal Sebah (psebah(AT)yahoo.fr), Aug 22 2002 Commented and edited by Daniel Forgues, Jul 28 2009 Commented and reference added by Jonathan Sondow, Nov 26 2010 Unsound terms after a(9) removed by Gord Palameta, Sep 06 2018 STATUS approved

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Last modified November 26 21:15 EST 2022. Contains 358362 sequences. (Running on oeis4.)