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 A353251 #12 Apr 09 2022 02:24:56
%S A353251 1,1,3,13,19,143,223,2521,4201,21563,37691,737161,1328521,31463413,
%T A353251 57821173,21404465,39854897,1267947073,2383173185,85428430547,
%U A353251 808549483039,1535039635999,2921975382559,128230606647497,245195521274057,2348840786785261,4508193056814061
%N A353251 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); denominators.
%H A353251 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>.
%H A353251 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LerchTranscendent.html">Lerch Transcendent</a>.
%H A353251 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_mean">Harmonic mean</a>.
%F A353251 a(n) = denominator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.
%e A353251 a(0) = 1,
%e A353251 a(1) = 2/(1/1 + 1/1) = 1,
%e A353251 a(2) = 2/(1/1 + 1/2) = 4/3,
%e A353251 a(3) = 2/(1/(4/3) + 1/3) = 24/13,
%e A353251 a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
%e A353251 This sequence gives the denominators: 1, 1, 3, 13, 19, ...
%t A353251 Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 26}] // Denominator (* or *)
%t A353251 a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 26}] // Denominator
%Y A353251 Cf. A353250 (numerators).
%Y A353251 Cf. A003149, A136128, A241519, A242376.
%K A353251 nonn,frac
%O A353251 0,3
%A A353251 _Vladimir Reshetnikov_, Apr 08 2022