A065418 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

6, 3, 5, 1, 6, 6, 3, 5, 4, 6, 0, 4, 2, 7, 1, 2, 0, 7, 2, 0, 6, 6, 9, 6, 5, 9, 1, 2, 7, 2, 5, 2, 2, 4, 1, 7, 3, 4, 2, 0, 6, 5, 6, 8, 7, 3, 3, 2, 3, 7, 2, 4, 5, 0, 8, 9, 9, 7, 3, 4, 4, 6, 0, 4, 8, 6, 7, 8, 4, 6, 1, 3, 1, 1, 6, 1, 3, 9, 1, 8, 8, 2, 0, 8, 0, 2, 9, 1, 3, 8, 6, 7, 6, 4, 0, 4, 6, 1, 7

Offset

0,1

Comments

For comparison: Product_{n>=5} (1-(3n-1)/(n-1)^3) = 3/8 . - R. J. Mathar, Feb 25 2009

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.

Links

Table of n, a(n) for n=0..98.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant C_1^(3).

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Formula

The constant equals Product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-A027376(n). - Michael Somos, Apr 05 2003

Example

0.635166354604271207206696591272522417342...

Mathematica

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{4, -3}, {-6, -24}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 17 2016 *)

Prog

(PARI) prodeulerrat(1-(3*p-1)/(p-1)^3, 1, 5) \\ Amiram Eldar, Mar 10 2021

Crossrefs

Cf. A066654, A065419, A027376.

Sequence in context: A195490 A195471 A305187 * A228725 A286982 A153595

Adjacent sequences: A065415 A065416 A065417 * A065419 A065420 A065421

Keyword

cons,nonn

Author

N. J. A. Sloane, Nov 15 2001

Status

approved