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%I #42 Feb 11 2023 12:20:34
%S 4,2,8,2,4,9,5,0,5,6,7,7,0,9,4,4,4,0,2,1,8,7,6,5,7,0,7,5,8,1,8,2,3,5,
%T 4,6,1,2,1,2,9,8,5,1,3,3,5,5,9,3,6,1,4,4,0,3,1,9,0,1,3,7,9,5,3,2,1,2,
%U 3,0,5,2,1,6,1,0,8,3,0,4,4,1,0,5,3,4,8,5,1,4,5,2,4,6,8,0,6,8,5,5,4,8,0,7,5,7
%N Decimal expansion of Product_{p prime}(1 - (2*p-1)/p^3).
%C Sum_{n <= x} A189021(n) ~ kx, where k is this constant. - _Charles R Greathouse IV_, Jan 24 2018
%C The probability that a number chosen at random is squarefree and coprime to another randomly chosen random (see Schroeder, 2009). - _Amiram Eldar_, May 23 2020, corrected Aug 04 2020
%D Manfred Schroeder, Number Theory in Science and Communication, 5th edition, Springer, 2009, page 59.
%H R. J. Mathar, <a href="https://arxiv.org/abs/0903.2514">Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers</a>, arXiv:0903.2514 [math.NT], 2009-2011; Equation (116).
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarefreeCouple.html">Carefree Couple</a>.
%F Equals A065463 divided by A013661. - _R. J. Mathar_, Mar 22 2011
%F Equals A065473 divided by A065480. - _R. J. Mathar_, May 02 2019
%F Equals (6/Pi^2)^2 * Product_{p prime} (1 + 1/(p^3 + p^2 - p - 1)) = 1.1587609... * (6/Pi^2)^2. - _Amiram Eldar_, May 23 2020
%F Equals lim_{m->oo} (1/m) * Sum_{k==1..m} (phi(k)/k)^2, where phi is the Euler totient function (A000010). - _Amiram Eldar_, Mar 12 2021
%e 0.428249505677094440218765707581823546...
%t $MaxExtraPrecision = 800; digits = 98; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms+10]]; r[n_Integer] := LR[[n]]; (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n-1]/(n-1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Apr 16 2016 *)
%o (PARI) prodeulerrat(1 - (2*p-1)/p^3) \\ _Amiram Eldar_, Mar 12 2021
%Y Cf. A000010, A057434, A078078, A056671.
%Y Cf. A065463, A013661.
%Y Cf. A065473, A065480.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
%E More digits from _Vaclav Kotesovec_, Dec 18 2019
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