This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369885 #12 Aug 05 2024 13:33:31
%S A369885 1,8,0,0,7,5,5,0,5,6,0,0,5,2,8,2,9,9,1,4,9,6,6,0,6,0,1,4,2,1,4,8,4,3,
%T A369885 1,8,1,4,4,5,6,6,3,7,8,3,8,1,8,4,1,7,9,3,0,2,7,1,8,6,6,7,5,9,1,7,2,9,
%U A369885 9,8,8,3,1,7,6,3,8,6,3,1,1,8,0,5,1,5,9,2,9,8,4,3,7,8,8,9,2,4,3,8,1,0,9,8,9
%N A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.
%H A369885 Bernard Candelpergher and Marc-Antoine Coppo, <a href="https://doi.org/10.1007/s11139-011-9361-7">A new class of identities involving Cauchy numbers, harmonic numbers and zeta values</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 305-328; <a href="https://math.univ-cotedazur.fr/~coppo/RamanujanJ%282012%29.pdf">alternative link</a>.
%H A369885 István Mező, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.121.07.648">Problem 11793</a>, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 7 (2014), p. 648; <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.6.613">A Series with Zetas</a>, Solution to Problem 11793 by FAU Problem Solving Group, ibid., Vol. 123, No. 6 (2016), p. 620.
%H A369885 Michael Ian Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">A catalog of the real numbers</a> (2011), p. 616.
%H A369885 Roberto Tauraso, <a href="https://www.mat.uniroma2.it/~tauraso/AMM/AMM11793.pdf">Problem 11793</a>.
%H A369885 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_number#Harmonic_numbers_for_real_and_complex_values">Harmonic number: Harmonic numbers for real and complex values</a>.
%F A369885 Equals Integral_{x>=1} H(x)/x^2 dx, where H(x) is the harmonic number for real variable x (Shamos, 2011).
%F A369885 Equals -zeta'(2) + Sum_{k>=3} (-1)^(k+1)*zeta(k)/(k-2) (Mező, 2014).
%F A369885 Equals Sum_{k>=1} lambda(k)*H(k)/(k^2*k!) + 1 + zeta(3) - gamma * zeta(2), where lambda(k) = abs(A006232(k)/A006233(k)) is the n-th non-alternating Cauchy number, H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma is Euler's constant (A001620) (Candelpergher and Coppo, 2012). - _Amiram Eldar_, Mar 18 2024
%e A369885 1.80075505600528299149660601421484318144566378381841...
%p A369885 evalf(sum((-1)^(k+1)*Zeta(k)/(k-2), k = 3 .. infinity) - Zeta(1, 2), 120)
%t A369885 RealDigits[NIntegrate[HarmonicNumber[x]/x^2, {x, 1, Infinity}, WorkingPrecision -> 120]][[1]]
%o A369885 (PARI) sumpos(k = 1, log(k+1)/k^2)
%o A369885 (PARI) sumalt(k = 3, (-1)^(k+1) * zeta(k)/(k-2)) - zeta'(2)
%Y A369885 Cf. A001008, A001620, A002805, A006232, A006233, A073002, A210593.
%K A369885 nonn,cons
%O A369885 1,2
%A A369885 _Amiram Eldar_, Feb 04 2024