This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A065421 #102 Apr 21 2025 16:03:37
%S A065421 1,9,0,2,1,6,0,5,8
%N 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.
%C A065421 The calculation of Brun's constant is "based on heuristic considerations about the distribution of twin primes" (Ribenboim, 1989).
%C A065421 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.
%C A065421 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
%D A065421 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 14.
%D A065421 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 133-135.
%D A065421 Paulo Ribenboim, The Book of Prime Number Records, 2nd. ed., Springer-Verlag, New York, 1989, p. 201.
%D A065421 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 193.
%H A065421 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), <a href="https://gallica.bnf.fr/ark:/12148/bpt6k96292009/f104.image">100-104</a> and <a href="https://gallica.bnf.fr/ark:/12148/bpt6k96292009/f128.image">124-128</a>.
%H A065421 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=BrunsConstant">Brun's constant</a>
%H A065421 Sebastian M. Cioabă and Werner Linde, <a href="https://bookstore.ams.org/view?ProductCode=AMSTEXT/58">A Bridge to Advanced Mathematics: from Natural to Complex Numbers</a>, Amer. Math. Soc. (2023) Vol. 58, see page 334.
%H A065421 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/brun/brun.html">Brun's Constant</a> [Broken link]
%H A065421 Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/brun/brun.html">Brun's Constant</a> [From the Wayback machine]
%H A065421 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/twins/twins.html">Enumeration to 10^14 of the twin primes and Brun's constant</a>, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
%H A065421 Thomas R. Nicely, <a href="/A001359/a001359.pdf">Enumeration to 10^14 of the twin primes and Brun's constant</a> [Local copy, pdf only]
%H A065421 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/counts.html">Prime Constellations Research Project</a>
%H A065421 P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Primes/twin.html">Numbers, constants and computation</a>
%H A065421 D. Shanks and J. W. Wrench, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352022-X">Brun's constant</a>, Math. Comp. 28 (1974) 293-299; 28 (1974) 1183; Math. Rev. 50 #4510.
%H A065421 H. Tronnolone, <a href="https://web.archive.org/web/20190317232307/http://www.maths.adelaide.edu.au/hayden.tronnolone/publications/pdfs/A_tale_of_two_primes.pdf">A tale of two primes</a>, COLAUMS Space, #3, 2013.
%H A065421 Wikipedia, <a href="http://en.wikipedia.org/wiki/Brun%27s_constant">Brun's constant</a>
%F A065421 Equals Sum_{n>=1} 1/A077800(n).
%F A065421 From _Dimitris Valianatos_, Dec 21 2013: (Start)
%F A065421 (1/5) + Sum_{n>=1, excluding twin primes 3,5,7,11,13,...} mu(n)/n =
%F A065421 (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)
%e A065421 (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... = 1.902160583209 +- 0.000000000781 [Nicely]
%Y A065421 Cf. A005597 (twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2)).
%Y A065421 Cf. A077800 (twin primes).
%K A065421 hard,more,nonn,cons,nice
%O A065421 1,2
%A A065421 _Robert G. Wilson v_, Sep 08 2000
%E A065421 Corrected by _N. J. A. Sloane_, Nov 16 2001
%E A065421 More terms computed by Pascal Sebah (pascal_sebah(AT)ds-fr.com), Jul 15 2001
%E A065421 Further terms computed by Pascal Sebah (psebah(AT)yahoo.fr), Aug 22 2002
%E A065421 Commented and edited by _Daniel Forgues_, Jul 28 2009
%E A065421 Commented and reference added by _Jonathan Sondow_, Nov 26 2010
%E A065421 Unsound terms after a(9) removed by _Gord Palameta_, Sep 06 2018