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

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