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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. 11
1, 9, 0, 2, 1, 6, 0, 5, 8, 3, 1, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
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.

REFERENCES

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.

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.

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, http://maths.adelaide.edu.au/hayden.tronnolone/publications/A_tale_of_two_primes.pdf

LINKS

Table of n, a(n) for n=1..13.

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

Wikipedia, Brun's constant

FORMULA

Comment from Dimitris Valianatos, Dec 21 2013: (1/5)  + Σ_{n=1 to ∞, excluding twin primes 3,5,7,11,13,...} μ(n)/n =

(1/5) + 1-1/2 +1/6+1/10+1/14+1/15+1/22-1/23+1/26-1/30+1/33+1/34+1/35-1/37+1/38+1/39-1/42... = 1.902160583...

EXAMPLE

1.902160583209 +- 0.000000000781 [Nicely]

CROSSREFS

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

Sequence in context: A221507 A089481 A188593 * A198556 A093767 A199175

Adjacent sequences:  A065418 A065419 A065420 * A065422 A065423 A065424

KEYWORD

hard,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

STATUS

approved

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Last modified August 1 08:04 EDT 2014. Contains 245112 sequences.