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 A117731 #43 Feb 16 2025 08:33:00
%S A117731 1,7,37,533,1627,18107,237371,95549,1632341,155685007,156188887,
%T A117731 3602044091,18051406831,54260455193,225175759291,13981692518567,
%U A117731 14000078506967,98115155543129,3634060848592973,3637485804655193
%N A117731 Numerator of the fraction n*Sum_{k=1..n} 1/(n+k).
%C A117731 a(n) almost always equals A082687(n), but differs for n in A125740.
%C A117731 p divides a((p-1)/3) for primes p in A002476, that is, primes of form 6*n + 1. - _Alexander Adamchuk_, Jul 16 2006
%H A117731 G. C. Greubel, <a href="/A117731/b117731.txt">Table of n, a(n) for n = 1..1000</a>
%H A117731 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%H A117731 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%F A117731 a(n) = numerator(n*Sum_{k=1..n} 1/(n+k)).
%F A117731 a(n) = numerator(n*(Psi(2*n+1) - Psi(n+1))).
%F A117731 a(n) = numerator(n*Sum_{k=1..2*n} (-1)^(k+1)/k).
%F A117731 a(n) = numerator(n*A058313(2*n)/A058312(2*n)).
%F A117731 a(n) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)), which is the numerator of the sum of all matrix elements of n X n Hilbert Matrix M(i,j) = 1/(i+j-1), (i,j = 1..n). The denominator is A117664(n). - _Alexander Adamchuk_, Apr 23 2006
%e A117731 The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/A296519.
%e A117731 For n=2, the n X n Hilbert matrix is
%e A117731 1 1/2
%e A117731 1/2 1/3
%e A117731 Thus, a(2) = numerator(1 + 1/2 + 1/2 + 1/3) = numerator(7/3) = 7.
%e A117731 The n X n Hilbert matrix begins as follows:
%e A117731 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e A117731 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e A117731 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e A117731 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e A117731 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e A117731 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%e A117731 ...
%t A117731 Numerator[Table[n Sum[1/(n + k), {k, n}], {n, 1, 100}]]
%t A117731 Numerator[Table[Sum[Sum[1/(i + j - 1), {i, n}], {j, n}], {n, 30}]] (* _Alexander Adamchuk_, Apr 23 2006 *)
%t A117731 Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 20}] // Numerator (* _Eric W. Weisstein_, Dec 14 2017 *)
%o A117731 (PARI) a(n) = numerator(n*sum(k=1, n, 1/(n+k))); \\ _Michel Marcus_, Dec 14 2017
%o A117731 (Magma) [Numerator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // _G. C. Greubel_, Jul 24 2023
%o A117731 (SageMath) [numerator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # _G. C. Greubel_, Jul 24 2023
%Y A117731 Cf. A296519 (denominators).
%Y A117731 Cf. A001008, A002476, A005249, A058313, A058312, A082687, A086881, A098118, A117664, A125740.
%K A117731 frac,nonn
%O A117731 1,2
%A A117731 _Alexander Adamchuk_, Apr 14 2006
%E A117731 Various sections edited by _Petros Hadjicostas_ and _Michel Marcus_, May 07 2020