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 A115563 #74 Feb 16 2025 08:33:00
%S A115563 2,1,0,9,7,4,2,8,0,1,2,3,6,8,9,1,9,7,4,4,7,9,2,5,7,1,9,7,6,1,6,5,5,1,
%T A115563 3,2,6,3,8,5,5,3,1,9,8,4,3,9,4,7,4,2,0,2,2,6,4,9,9,1,5,6,0,3,1,9,2,8,
%U A115563 1,4,6,9,4,9,3,9,1,3,6,8,7,4,1,7,7,1,6,9,2,9,1,3,7,7,1,8,6,2,3,2,1,3,5,8,3,8,7,6,6,5,3,4,7,2,6,0,9,7,3,8,9,0,3,5,7,7,9,5,0,8,6,5,9,4,8,9,4,2,4,6,5
%N A115563 Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).
%C A115563 Sum_{n>1} 1/(n*log(n)^2) is a tiny bit greater than (zeta(2))^(3/2) = (Pi^2 / 6)^(3/2) = 2.109709908063657.... - _Daniel Forgues_, Mar 30 2012
%C A115563 From _Bernard Schott_, Oct 03 2021: (Start)
%C A115563 Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 3, 4, 5 see respectively A145419, A145420, A145421).
%C A115563 As H(n) ~ log(n), compare with A347145. (End)
%H A115563 John V. Baxley, <a href="http://www.jstor.org/stable/2691241">Euler's constant, Taylor's formula, and slowly converging series</a>, Math. Mag. 65 (1992), 302-313.
%H A115563 Bart Braden, <a href="http://www.jstor.org/stable/2324995">Calculating sums of infinite series</a>, Am. Math. Monthly 99 (1992) 649-655.
%H A115563 David Broadhurst, <a href="http://groups.yahoo.com/group/primeform/message/7081">Re: need help about 2 constants</a>, primeforum, Mar 10 2006.
%H A115563 Pierre CAMI, David Broadhurst, <a href="/A115563/a115563.txt">Need help about 2 constants</a>, digest of 3 messages in primeform Yahoo group, Mar 10, 2006. [Cached copy]
%H A115563 Rick Kreminski, <a href="http://www.jstor.org/stable/2687066">Using Simpson's rule to approximate sums of infinite series</a>, College Math. J. 28 (1997), 368-376.
%H A115563 R. J. Mathar, <a href="http://arxiv.org/abs/0902.0789">The series limit of sum_k 1/[k*log k *(log log k)^2]</a>, arXiv:0902.0789 [math.NA], 2009, App. A.
%H A115563 S.O.S. Math, <a href="http://www.sosmath.com/calculus/series/bertrand/bertrand.html">Bertrand Series</a>.
%H A115563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConvergentSeries.html">Convergent Series</a>
%H A115563 Wikipédia, <a href="https://fr.wikipedia.org/wiki/Série_de_Bertrand">Série de Bertrand</a> (in French).
%e A115563 2.10974280123689197447925719761655132638553198439474202264991560319281...
%t A115563 digits = 150; NSum[1/(n*Log[n]^2), {n, 2, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] (* _Vaclav Kotesovec_, Mar 01 2016, after _Jean-François Alcover_ *)
%t A115563 maxiter = 20; nn = 10000; alfa = 2; bas = Sum[1/(k*Log[k]^alfa), {k, 2, nn}] + 1/((alfa - 1)*Log[nn + 1/2]^(alfa - 1)); sub = 0; Do[sub = sub + 1/4^s/(2*s + 1)! * NSum[(D[1/(x*Log[x]^alfa), {x, 2 s}]) /. x -> k, {k, nn + 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]; Print[bas - sub], {s, 1, maxiter}] (* _Vaclav Kotesovec_, Jun 12 2022 *)
%Y A115563 Cf. A145419, A145420, A145421, A363632, A363633.
%Y A115563 Cf. A137245, A257812. A097906 is a similar sum.
%Y A115563 Cf. A118582, A347145.
%K A115563 cons,nonn
%O A115563 1,1
%A A115563 _Pierre CAMI_, Mar 11 2006
%E A115563 Removed incorrect speculations about relations to A097906 - _R. J. Mathar_, Oct 14 2010
%E A115563 More terms from _Robert G. Wilson v_, Dec 12 2012
%E A115563 Corrected a(55) and beyond, _Vaclav Kotesovec_, Mar 01 2016