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.

1, 9, 0, 2, 1, 6, 0, 5, 8

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

Table of n, a(n) for n=1..9.

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

Sebastian M. Cioabă and Werner Linde, A Bridge to Advanced Mathematics: from Natural to Complex Numbers, Amer. Math. Soc. (2023) Vol. 58, see page 334.

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: A370347 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