

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.


26




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 [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. 133135.
P. Ribenboim, The Book of Prime Number Records, 2nd. ed., SpringerVerlag, 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), 100104 and 124128.
D. Shanks and J. W. Wrench, Brun's constant, Math. Comp. 28 (1974) 293299; 28 (1974) 1183; Math. Rev. 50 #4510.


FORMULA

(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)).


KEYWORD



AUTHOR



EXTENSIONS

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


STATUS

approved



