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 A101627 #25 Mar 01 2022 07:25:58
%S A101627 1,39,241,34883,14039,1516871,7601151,875425657,7887002813,
%T A101627 7095769757767,14199583385459,75087685321529,75113436870869,
%U A101627 927229349730873529,927436191807263569,305182576081725442901,23479178371879154033,37713848011377144613,37717984058802320713,135759786815564675620247
%N A101627 Numerator of partial sums of a certain series.
%C A101627 The denominators are given in A101628.
%C A101627 Second member (m=3) of a family defined in A101028.
%C A101627 The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 8*sum(zeta(2*k+1)/3^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 12*(log(3)-1) = 1.18334746...; see the Abramowitz-Stegun (given in A101028) reference p. 259, eq. 6.3.15 with z=1/3 together with p. 258.
%H A101627 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A101627 Wolfdieter Lang, <a href="/A101627/a101627.txt">Rationals s(n) and more.</a>
%F A101627 a(n)=numerator(s(n)) with s(n)=24*sum(1/((3*k-1)*(3*k)*(3*k+1)), k=1..n).
%e A101627 s(3)= 24*(1/(2*3*4)+ 1/(5*6*7) + 1/(8*9*10)) = 241/210, hence a(3)=241 and A101628(3)=210.
%t A101627 Numerator[Accumulate[Table[8/(9k^3-k),{k,20}]]]
%o A101627 (PARI) a(n) = numerator(24*sum(k=1, n, 1/((3*k-1)*(3*k)*(3*k+1))));
%Y A101627 Cf. A101028 (m=2), A101629 (m=4), A101631 (m=5).
%Y A101627 Cf. A101628 (denominators).
%K A101627 nonn,frac,easy
%O A101627 1,2
%A A101627 _Wolfdieter Lang_, Dec 23 2004
%E A101627 More terms from _Michel Marcus_, Mar 01 2022