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

Author

N. J. A. Sloane, Nov 15 2001

Keywords

• cons
• nonn

Comments

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

Examples

			0.635166354604271207206696591272522417342...
		

References

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

Links

• 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).
• B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Crossrefs

Cf. A066654, A065419, A027376.

Programs

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 *)

PARI

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

Formula

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