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A386712 Decimal expansion of Sum_{k>=2} (zeta(k)-1)/(k+2).

%I A386712 #7 Jul 31 2025 06:59:03
%S A386712 2,2,4,4,8,0,6,2,4,4,2,7,2,4,7,7,7,9,5,8,9,6,6,0,2,4,6,4,1,4,6,8,6,9,
%T A386712 3,0,9,2,9,8,0,9,9,8,7,0,4,5,1,7,1,8,2,0,2,4,7,8,8,1,4,3,5,1,7,4,2,2,
%U A386712 5,6,6,2,4,8,0,3,6,3,6,9,9,8,0,7,2,2,4,1,4,6,2,6,8,4,4,6,0,4,1,4,6,3,0,2,9
%N A386712 Decimal expansion of Sum_{k>=2} (zeta(k)-1)/(k+2).
%D A386712 Hari M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012. See eq. (543), p. 320.
%H A386712 Ovidiu Furdui, <a href="https://doi.org/10.1080/10652469.2015.1031129">The evaluation of a class of fractional part integrals</a>, Integral Transforms and Special Functions, Vol. 26, No. 8 (2015), pp. 635-641.
%H A386712 Michael I. Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">Shamos's catalog of the real numbers</a>, 2011. See p. 290.
%H A386712 Hari M. Srivastava and Junesang Choi, <a href="https://link.springer.com/book/9789048157280">Series Associated with the Zeta and Related Functions</a>, Springer Science+Business Media Dordrecht, 2001. See eq. (517), p. 219.
%F A386712 Equals 11/6 - gamma/3 - 2*log(A) - log(2*Pi)/2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant (Srivastava and Choi, 2001).
%F A386712 Equals -Sum_{k>=2} (k^2*log(1-1/k) + k + 1/(3*k) + 1/2) (Shamos, 2011).
%e A386712 0.22448062442724777958966024641468693092980998704517...
%t A386712 RealDigits[11/6 - EulerGamma/3 - 2*Log[Glaisher] - Log[2*Pi]/2, 10, 120][[1]]
%o A386712 (PARI) 11/6 - Euler/3 - 2*(1/12-zeta'(-1)) - log(2*Pi)/2
%Y A386712 Cf. A001620 (gamma), A061444, A074962 (A), A225746.
%Y A386712 Sum_{k>=2} (zeta(k)-1)/(k+m): A153810 (m=0), A386711 (m=1), this constant (m=2).
%K A386712 nonn,cons
%O A386712 0,1
%A A386712 _Amiram Eldar_, Jul 31 2025