A061642 Decimal expansion of Hardy-Littlewood constant for prime quadruples.

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

Offset

1,1

Comments

Computed by Robert Harley.

References

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Links

Table of n, a(n) for n=1..105.

Steven R. Finch, Hardy-Littlewood Constants. [Broken link]

Steven R. Finch, Hardy-Littlewood Constants. [From the Wayback machine]

Warut Roonguthai, Large Prime Quadruplets, NMBRTHRY Archives.

Eric Weisstein's World of Mathematics, Prime Quadruplet.

Formula

Equals (27/2) * Product_{p prime > 3} (p^3)*(p-4)/((p-1)^4) using 27/2 = (3*(11+13)+(17+19))/4. - Frank Ellermann, Mar 31 2020

Example

4.151180863237415757165285561959537515799410019333963032027163...

Mathematica

$MaxExtraPrecision = 1500; digits = 105; terms = 1500; 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]]; (27/2)* 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 16 2016 *)

Prog

(PARI) (27/2) * prodeulerrat((p^3)*(p-4)/((p-1)^4), 1, 5) \\ Amiram Eldar, Mar 12 2021

Crossrefs

Cf. A065419 (constant without factor 27/2), A333586, A333587.

Sequence in context: A185373 A244759 A194127 * A143313 A132588 A195986

Adjacent sequences: A061639 A061640 A061641 * A061643 A061644 A061645

Keyword

cons,nonn

Author

Jason Earls, Jun 13 2001

Status

approved