

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.


14




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 HardyLittlewood 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 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

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) 100104 and 124128.
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 14.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 133135.
P. Ribenboim, The Book of Prime Number Records, 2nd. ed., SpringerVerlag, New York, 1989, p. 201.


LINKS

Table of n, a(n) for n=1..9.
C. K. Caldwell, The Prime Glossary, Brun's constant
S. R. Finch, Brun's Constant
T. R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant.
T. 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) 293299; 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 } (11/(p1)^2).
Sequence in context: A221507 A089481 A188593 * A198556 A261169 A093767
Adjacent sequences: A065418 A065419 A065420 * A065422 A065423 A065424


KEYWORD

hard,more,nonn,cons,nice,changed


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)dsfr.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



