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 A065421 Decimal expansion of Viggo Brun's constant B, also known as the twin primes Brun's constant B_2: Sum (1/p + 1/q) as (p,q) runs through the twin primes. 10

%I

%S 1,9,0,2,1,6,0,5,8,3,1,0,4

%N Decimal expansion of Viggo Brun's constant B, also known as the twin primes Brun's constant B_2: Sum (1/p + 1/q) as (p,q) runs through the twin primes.

%C The calculation of Brun's constant is “based on heuristic considerations about the distribution of twin primes” (Ribenboim, 1989).

%C 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.

%D V. Brun, La serie 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... ou les denominateurs sont "nombres premiers jumeaux" est convergente ou finie, Bull Sci. Math. 43 (1919) 100-104 and 124-128.

%D R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 14.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 133-135.

%D P. Ribenboim, The Book of Prime Number Records, 2nd. ed., Springer- Verlag, New York, 1989, p. 201.

%D D. Shanks and J. W. Wrench, Brun's constant, Math. Comp. 28 (1974) 293-299; 28 (1974) 1183; Math. Rev. 50 #4510.

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=BrunsConstant">Brun's constant</a>

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/brun/brun.html">Brun's Constant</a>

%H P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Primes/twin.html">Numbers, constants and computation</a>

%H T. R. Nicely, <a href="http://www.trnicely.net/twins/twins.html">Enumeration to 10^14 of the twin primes and Brun's constant</a>.

%H T. R. Nicely, <a href="http://www.trnicely.net/counts.html">Prime Constellations Research Project</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Brun%27s_constant">Brun's constant</a>

%F (1/3+1/5) + (1/5+1/7) + (1/11+1/13) + ...

%e 1.902160583209 +- 0.000000000781 [Nicely]

%Y Cf. A005597 Decimal expansion of twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

%K hard,nonn,cons,nice

%O 1,2

%A _Robert G. Wilson v_, Sep 08 2000

%E Corrected by _N. J. A. Sloane_, Nov 16, 2001

%E More terms computed by Pascal Sebah (pascal_sebah(AT)ds-fr.com), Jul 15, 2001

%E Further terms computed by Pascal Sebah (psebah(AT)yahoo.fr), Aug 22 2002

%E Commented and edited by _Daniel Forgues_, Jul 28 2009

%E Commented and reference added by _Jonathan Sondow_, Nov 26 2010

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Last modified May 23 07:06 EDT 2013. Contains 225585 sequences.