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A246749 Decimal expansion of F'(rho), an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number, where the function F and the constant rho are defined in A246746.

%I A246749 #10 Jan 17 2020 16:04:00
%S A246749 5,6,9,7,7,5,8,9,3,4,2,3,0,1,9,2,6,7,5,7,5,2,9,1,3,7,0,4,6,8,5,2,4,7,
%T A246749 8,9,7,8,5,8,1,0,1,9,8,2,1,7,8,3,5,7,3,5,9,3,4,5,9,5,6,7,1,7,5,8,4,1,
%U A246749 1,4,4,0,5,3,8,6,6,0,6,7,7,6,8,3,1,7,8,4,7,5,1,5,7,4,3,8,9,2,8,8,5
%N A246749 Decimal expansion of F'(rho), an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number, where the function F and the constant rho are defined in A246746.
%H A246749 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 16.
%H A246749 Kevin Ford, <a href="http://www.math.uiuc.edu/~ford/wwwpapers/totients.pdf">The distribution of Totients</a>
%F A246749 Let F(x) = sum_{k >= 1} ((k+1)*log(k+1) - k*log(k) - 1)*x^k.
%F A246749 F'(rho), where rho is the unique solution on [0,1) of F(rho)=1,
%e A246749 5.6977589342301926757529137046852478978581019821783573593459567...
%t A246749 digits = 101; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; F'[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*k*x^(k - 1), {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; RealDigits[F'[rho], 10, digits] // First
%Y A246749 Cf. A246746.
%K A246749 nonn,cons
%O A246749 1,1
%A A246749 _Jean-François Alcover_, Sep 02 2014