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%I #66 May 24 2023 05:52:11
%S 8,8,1,5,1,3,8,3,9,7,2,5,1,7,0,7,7,6,9,2,8,3,9,1,8,2,2,9,0,3,2,2,7,8,
%T 4,7,1,2,9,8,6,9,2,5,7,2,0,8,0,7,6,7,3,3,6,7,0,1,6,8,5,3,5,5,4,8,6,5,
%U 7,9,0,6,3,7,9,4,1,6,9,7,4,1,0,2,2,0,4,5,5,1,7,9,7,0,2,0,9,6
%N Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).
%C From _Richard R. Forberg_, May 22 2023: (Start)
%C This constant is the asymptotic mean of (phi(n)/n)*(sigma(n)/n), where phi is the Euler totient function (A000010) and sigma is the sum-of-divisors function (A000203).
%C In contrast, the product of the separate means, mean(phi(n)/n) * mean(sigma(n)/n), converges to 1, with the asymptotic mean(sigma(n)/n) = Pi^2/6 = zeta(2). See A013661.
%C Also see A062354. (End)
%H Eckford Cohen, <a href="http://dx.doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a>, page 7. [Cached copy, with permission of the author]
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50 and 85.
%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers</a>, arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(2).
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/881513.html">Generalized expansions of real numbers</a>, 2006.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuadraticClassNumberConstant.html">Quadratic Class Number Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%F Sum_{n>=1} phi(n)/(n*J(n)) = (this constant)*A013661 with phi()=A000010() and J() = A007434() [Cohen, Corollary 5.1.1]. - _R. J. Mathar_, Apr 11 2011
%e 0.88151383972517077692839182290...
%t $MaxExtraPrecision = 1000; digits = 98; terms = 1000; LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*PrimeZetaP[n-1]/(n-1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Apr 16 2016 *)
%o (PARI) prodeulerrat(1 - 1/(p^2*(p+1))) \\ _Amiram Eldar_, Mar 14 2021
%Y Cf. A000010, A007434, A013661, A062354, A078087.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
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