A065472 Decimal expansion of Product_{p prime} (1 - 1/(p+1)^2).

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

Offset: 0

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• cons
• nonn

Comments

The probablity that two randomly chosen squarefree numbers are coprime. - Amiram Eldar, Aug 04 2020

The asymptotic mean of A001157(n)/(n*A000203(n)). - Richard R. Forberg, May 27 2023

Examples

			0.7758835100038954996204042844279...
		

Links

• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica, Vol. 40, No. 1 (1989), pp. 19-30.

Crossrefs

Cf. A000203, A001157, A008683, A038063, A078091, A116393, A145388.

Programs

Mathematica

digits = 98; Exp[NSum[(-1)^n*(2^(n-1)-2)*PrimeZetaP[n-1]/(n-1), {n, 3, Infinity}, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

PARI

prodeulerrat(1 - 1/(p+1)^2) \\ Amiram Eldar, Mar 17 2021

Formula

Equals lim_{n->oo} (Pi^2/(3*n^2*log(n))) * Sum_{k=1..n} A145388(k). - Amiram Eldar, May 14 2019

Equals Sum_{k>=1} mu(k)/sigma(k)^2, where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022

Extensions

Definition corrected by Dan Asimov, Apr 15 2006