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
Examples
4.151180863237415757165285561959537515799410019333963032027163...
References
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
Links
- 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.
Programs
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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 *) -
PARI
(27/2) * prodeulerrat((p^3)*(p-4)/((p-1)^4), 1, 5) \\ Amiram Eldar, Mar 12 2021
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
Comments