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%I #64 Apr 29 2023 08:09:12
%S 3,2,2,6,3,4,0,9,8,9,3,9,2,4,4,6,7,0,5,7,9,5,3,1,6,9,2,5,4,8,2,3,7,0,
%T 6,6,5,7,0,9,5,0,5,7,9,6,6,5,8,3,2,7,0,9,9,6,1,8,1,1,2,5,2,4,5,3,2,5,
%U 0,0,6,3,4,8,6,2,4,4,6,0,9,8,8,4,5,2,3,4,8,1,5,6,8,5,6,3,7,5,5,2,1,7,7,2,7,3
%N Decimal expansion of Product_{p prime} (1 - 2/p^2).
%C Density of A007674, squarefree n such that n + 1 is squarefree. - _Charles R Greathouse IV_, Aug 10 2011
%C Product_{k>=1} (1 - 2/k^2) = sin(sqrt(2)*Pi) / (sqrt(2)*Pi). - _Vaclav Kotesovec_, May 23 2020
%C The asymptotic probability that, for two integers k and m, 0 < k <= m, we have gcd(k*(k+1), m) = 1 (when k and m are chosen at random in the range 1..n and n->oo) (Tóth and Sándor, 1989). - _Amiram Eldar_, Apr 29 2023
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High precision computation of Hardy-Littlewood constants</a>, (1998).
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%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. 163 and 184.
%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, constant F_1^(2).
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H L. Tóth and J. Sándor, <a href="https://www.fq.math.ca/Scanned/27-2/toth.pdf">An asymptotic formula concerning a generalized Euler function</a>, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 176-180.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%H Zack Wolske, <a href="http://blog.math.toronto.edu/GraduateBlog/files/2018/05/ZWolskePhDThesisJune14-1.pdf">Number Fields with Large Minimal Index</a>, Ph.D. Thesis, University of Toronto, 2018.
%e 0.322634098939244670579531692548...
%t $MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{0, 2}, {-4, 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 *)
%o (PARI) prodeulerrat(1 - 2/p^2) \\ _Amiram Eldar_, Mar 16 2021
%Y Cf. A007674, A058026, A065493, A074893.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
%E Edited by _Dean Hickerson_, Sep 10 2002
%E More digits from _Vaclav Kotesovec_, Dec 18 2019
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