cp's OEIS Frontend

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

%I A363540 #5 Jun 09 2023 01:30:15
%S A363540 5,8,2,1,7,4,0,0,8,5,0,4,8,6,4,6,5,2,8,8,9,6,8,6,8,6,1,5,5,0,2,0,4,1,
%T A363540 3,4,3,1,5,0,3,3,3,2,4,3,1,9,5,7,7,0,1,1,4,4,2,4,0,3,9,2,7,6,4,7,6,4,
%U A363540 1,3,9,7,2,2,5,9,8,1,8,9,7,4,9,5,1,8,9,0,4,2,8,5,7,4,3,2,3,1,9,0,9,6,5,9,7
%N A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).
%H A363540 Iaroslav V. Blagouchine, <a href="https://doi.org/10.1016/j.jnt.2015.06.012">Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only</a>, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.
%F A363540 Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).
%e A363540 5.82174008504864652889686861550204134315033324319577...
%t A363540 RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]
%Y A363540 Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.
%K A363540 nonn,cons
%O A363540 1,1
%A A363540 _Amiram Eldar_, Jun 09 2023