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A247038 Decimal expansion of Integral_{x=0..1} log(floor(1/x))/(1+x) dx.

%I A247038 #27 Feb 16 2025 08:33:23
%S A247038 6,8,4,7,2,4,7,8,8,5,6,3,1,5,7,1,2,3,2,9,9,1,4,6,1,4,8,7,5,5,7,7,7,6,
%T A247038 2,0,4,6,0,6,7,5,4,1,6,3,3,7,4,4,8,8,3,6,6,0,6,2,8,9,8,6,7,8,1,5,9,5,
%U A247038 6,8,8,2,1,7,6,2,6,9,3,6,1,0,4,3,7,0,7,6,8,1,4,3,4,9,5,8,5,8,1,0,0,9,9,7
%N A247038 Decimal expansion of Integral_{x=0..1} log(floor(1/x))/(1+x) dx.
%C A247038 The same integral with 1/x instead of floor(1/x) evaluates to Pi^2/12 = A072691.
%D A247038 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8 Khinchin-Lévy constants, p. 61.
%H A247038 David Bailey, Jonathan Borwein and Richard Crandall, <a href="https://doi.org/10.1090/S0025-5718-97-00800-4">On the Khintchine constant</a>, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 417-431.
%H A247038 Daniel Shanks and John W. Wrench, Jr., <a href="https://www.jstor.org/stable/2309633">Khintchine's constant</a>, The American Mathematical Monthly, Vol. 66, No. 4 (1959), pp. 276-279.
%H A247038 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/KhinchinsConstant.html">Khinchin's Constant</a>
%F A247038 Equals log(2)*log(K), where K is Khinchin's constant A002210 = 2.685452...
%F A247038 From _Amiram Eldar_, Aug 19 2020: (Start)
%F A247038 Equals Sum_{k>=1} (zeta(2*k)-1)/k * (1 - 1/2 + 1/3 - ... + 1/(2*k - 1)).
%F A247038 Equals -Sum_{k>=2} log(1-1/k) * log(1+1/k). (End)
%e A247038 0.6847247885631571232991461487557776204606754163374488366...
%t A247038 RealDigits[Log[2]*Log[Khinchin], 10, 104] // First
%o A247038 (Python)
%o A247038 from mpmath import mp, log, khinchin
%o A247038 mp.dps=106
%o A247038 print([int(n) for n in list(str(log(2)*log(khinchin)))[2:-2]]) # _Indranil Ghosh_, Jul 08 2017
%Y A247038 Cf. A002210, A072691.
%K A247038 nonn,cons,easy
%O A247038 0,1
%A A247038 _Jean-François Alcover_, Sep 10 2014