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A273556 Decimal expansion of Rosser's constant. 0
8, 3, 2, 4, 2, 9, 0, 6, 5, 6, 6, 1, 9, 4, 5, 2, 7, 8, 0, 3, 0, 8, 0, 5, 9, 4, 3, 5, 3, 1, 4, 6, 5, 5, 7, 5, 0, 4, 5, 4, 4, 5, 3, 1, 8, 0, 7, 7, 4, 1, 7, 0, 5, 3, 2, 4, 0, 8, 9, 3, 9, 9, 1, 2, 9, 6, 0, 3, 4, 7, 0, 7, 1, 3, 9, 4, 8, 1, 1, 4, 2, 4, 2, 1, 9, 1, 6, 2, 7, 2, 2, 5, 0, 4, 6, 3, 8, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - Amiram Eldar, Jun 20 2021

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

LINKS

Table of n, a(n) for n=0..97.

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).

Eric Weisstein's World of Mathematics, Twin Primes Constant.

FORMULA

4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.

Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.

EXAMPLE

0.832429065661945278030805943531465575045445318077417053240893991296...

MATHEMATICA

digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];

RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First

PROG

(PARI) 4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 17 2021

CROSSREFS

Cf. A005597, A091724, A246061.

Sequence in context: A198843 A346718 A119277 * A346713 A154014 A063568

Adjacent sequences: A273553 A273554 A273555 * A273557 A273558 A273559

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, May 25 2016

STATUS

approved

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Last modified December 6 15:49 EST 2022. Contains 358644 sequences. (Running on oeis4.)