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 A168218 #26 Jun 19 2024 11:26:53
%S A168218 6,0,5,5,2,1,7,8,8,8,8,2,6,0,0,4,4,7,6,9,9,5,4,9,0,0,5,2,0,7,2,4,0,4,
%T A168218 4,7,3,0,3,2,3,8,8,9,8,4,5,5,0,6,5,7,8,3,3,1,1,1,4,7,5,9,0,4,2,0,6,8,
%U A168218 9,4,1,1,9,7,8,0,8,8,6,8,1,7,6,1,1,8,3,1,2,8,4,1,9,3,0,8,9,4,0,1,9,8,6,9,9
%N A168218 Decimal expansion of the Sum_{k=2..infinity} 1/(k^2*log(k)).
%C A168218 Also the value of the integral of the fractional part of the Riemann zeta function from 2 to infinity. - _Alexander Bock_, Apr 01 2014
%H A168218 R. P. Boas, Jr, <a href="http://www.jstor.org/stable/2318865">Partial sums of Infinite Series, and How They Grow</a>, Am. Math. Monthly 84 (4) (1977) 237-235 [<a href="http://www.ams.org/mathscinet-getitem?mr=440240">MR0440240</a>].
%F A168218 Equals Integral_{x>=2} (zeta(x) - 1) dx.
%e A168218 equals 1/(4*log(2))+1/(9*log(3))+1/(16*log(4))+ .... + = 0.605521788882600447699549005207240447303238898..
%t A168218 (* Computation needs a few minutes for 105 digits *) digits = 105; NSum[ 1/(n^2*Log[n]), {n, 2, Infinity}, NSumTerms -> 500000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 12}}] // RealDigits[#, 10, digits] & // First (* _Jean-François Alcover_, Feb 11 2013 *)
%t A168218 RealDigits[NIntegrate[Zeta[x] - 1, {x, 2, Infinity}, WorkingPrecision->110], 10, 100] (* _Alexander Bock_, Apr 01 2014 *)
%o A168218 (PARI) intnum(x=2,[oo,log(2)],zeta(x)-1) \\ _Charles R Greathouse IV_, Apr 01 2014
%o A168218 (PARI) suminf(k=2,1/k^2/log(k)) \\ _Charles R Greathouse IV_, Apr 01 2014
%Y A168218 Cf. A099769, A115563, A257812.
%K A168218 cons,nonn
%O A168218 0,1
%A A168218 _R. J. Mathar_, Nov 20 2009
%E A168218 More terms from _Jean-François Alcover_, Feb 11 2013