A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6*p^2-4*p+1)/(p-1)^4).

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

Offset

0,1

Comments

For comparison: Product_{n>=5} (1-(6n^2-4n+1)/(n-1)^4) = 3/32. - 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^(4).

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

Example

0.30749487875832709312335448607107685302...

Mathematica

$MaxExtraPrecision = 1000; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{5, -4}, {-12, -60}, 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-(6*p^2-4*p+1)/(p-1)^4, 1, 5) \\ Amiram Eldar, Mar 10 2021

Crossrefs

Cf. A065418, A027377, A065431.

Sequence in context: A369094 A322018 A200339 * A019092 A115870 A115860

Adjacent sequences: A065416 A065417 A065418 * A065420 A065421 A065422

Keyword

cons,nonn

Author

N. J. A. Sloane, Nov 15 2001

Extensions

A sign in the definition corrected by R. J. Mathar, Feb 25 2009

Status

approved