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 A125194 #6 Feb 16 2025 08:33:04
%S A125194 41,1599366601,10877829357646990581304675244472669289,
%T A125194 100935935338172297894217692920950359818733561,
%U A125194 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441
%N A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.
%C A125194 Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3 (see A120290(n)). The numerator of generalized harmonic number H((p-1)/2,2p) is divisible by p^2 for prime p>3.
%H A125194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A125194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%F A125194 a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,(Prime[n]-1)/2} ]] / Prime[n]^2 for n>2.
%e A125194 Prime[3] = 5.
%e A125194 a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.
%t A125194 Do[p=Prime[k];f=0;Do[f=f+1/n^(2p);g=Numerator[f];If[IntegerQ[g/(p)^2],Print[{p,g/p^2}]],{n,1,(p-1)/2}],{k,1,100}]
%Y A125194 Cf. A120290, A119722, A001008, A007406, A007408, A007410.
%K A125194 frac,nonn
%O A125194 3,1
%A A125194 _Alexander Adamchuk_, Jan 13 2007