A065469 Decimal expansion of Product_{p prime} (1 - 1/(p^2-1)).

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

Offset: 0

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• cons
• nonn

Examples

			0.53071182047204479497294377247...
		

Links

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

Crossrefs

Cf. A000010, A000203, A007434, A008683, A013661, A065474, A072101.

Programs

Mathematica

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = LinearRecurrence[{0, 3, 0, -2}, {0, 0, -2, 0}, 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 18 2016 *)

PARI

prodeulerrat(1 - 1/(p^2-1)) \\ Amiram Eldar, Mar 13 2021

Formula

Product of A013661 by A065474. - R. J. Mathar, Mar 26 2011

From Amiram Eldar, Jan 14 2022: (Start)

Equals Sum_{k>=1} mu(k)/(phi(k)*sigma(k)), where mu is the Möbius function (A008683), phi is the Euler totient function (A000010) and sigma(k) is the sum of divisors of k (A000203).

Equals Sum_{k>=1} mu(k)/J_2(k), where J_2 is Jordan's totient function (A007434). (End)