|
Search: id:A005597
|
|
|
| A005597 |
|
Decimal expansion of twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2).
(Formerly M4056)
|
|
+0
9
|
|
| 6, 6, 0, 1, 6, 1, 8, 1, 5, 8, 4, 6, 8, 6, 9, 5, 7, 3, 9, 2, 7, 8, 1, 2, 1, 1, 0, 0, 1, 4, 5, 5, 5, 7, 7, 8, 4, 3, 2, 6, 2, 3, 3, 6, 0, 2, 8, 4, 7, 3, 3, 4, 1, 3, 3, 1, 9, 4, 4, 8, 4, 2, 3, 3, 3, 5, 4, 0, 5, 6, 4, 2, 3, 0, 4, 4, 9, 5, 2, 7, 7, 1, 4, 3, 7, 6, 0, 0, 3, 1, 4, 1, 3, 8, 3, 9, 8, 6, 7, 9, 1, 1, 7, 7, 9
(list; cons; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
Occurs as part of a conjectured formula for the density of the twin primes.
|
|
REFERENCES
|
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.
J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.
Philippe Flajolet and Ilan Vardi, Zeta function Expansions of Classical constants, Feb. 18, 1996
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 84-93
|
|
LINKS
|
C. K. Caldwell, The Prime Glossary, twin prime constant
G. Niklasch, Some number theoretical constants: 1000-digit values
G. Niklasch, Twin primes constant
Pascal Sebah (pascal_sebah(AT)ds-fr.com), Numbers, constants and computation (gives 5000 digits)
S. Plouffe, The twin primes constant
S. Plouffe, Plouffe's Inverter, The twin primes constant
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
|
|
FORMULA
|
prod(k>=2, (zeta(k)*(1-1/2^k))^(-sum(d/k, mu(d)*2^(k/d))/k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2003
|
|
EXAMPLE
|
0.6601618158468695739278121100145557784326233602847334133194484233354056423...
|
|
PROGRAM
|
(PARI) ?\p1000 ? 175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))
|
|
CROSSREFS
|
Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271.
See also A065421 for another constant with the same name.
Sequence in context: A006806 A002892 A055667 this_sequence A081825 A028969 A029681
Adjacent sequences: A005594 A005595 A005596 this_sequence A005598 A005599 A005600
|
|
KEYWORD
|
cons,nonn,nice
|
|
AUTHOR
|
njas
|
|
EXTENSIONS
|
More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 08 2001
|
|
|
Search completed in 0.002 seconds
|
