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 A027459 #42 Feb 16 2025 08:32:35
%S A027459 1,7,85,415,12019,13489,726301,3144919,30300391,32160403,4102360483,
%T A027459 4301068993,758647585777,112686856171,3336876977,96568406789,
%U A027459 28776062218037,29608882035581,1568274265798307,11256448518043769
%N A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.
%C A027459 Originally defined as the first column of A027447, but now contains numerator in reduced form (cf. A329108). - _Sean A. Irvine_, Nov 04 2019
%C A027459 Numerators of the binomial transform of (-1)^n/(n+1)^3. The matrix a[i,j] below is the product of the binomial matrix and the matrix with general term binomial(i,j)(-1)^(i-j)/(i+1)^3. - _Paul Barry_, Aug 06 2004
%C A027459 From _Alexander Adamchuk_, Jan 02 2007 [edited by _Jon E. Schoenfield_, Mar 08 2015]: (Start)
%C A027459 Also a(n) is a numerator of S(n) = Sum_{k=1..n} H(k)/k, where H(k) is the k-th harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k).
%C A027459 S(n) = Sum_{k=1..n} H(k)/k = 1/2*(H(n)^2 + H(n,2)), where H(n,2) = Sum_{i=1..n} 1/i^2 = A007406(n)/A007407(n).
%C A027459 p divides a(p-1) and a(p-2) for prime p>3. a(n) is prime for n = {2, 7, 26, 31, 43, 53, 68, 80, 91, 123, 175, 236, 458, ...}. (End)
%C A027459 The n-fold repeated integral of (1/2)*log(x)^2 (all improper integrals with the lower limits of integration equal to 0) = x^n/n! * ( (1/2)*log(x)^2 - H(n)*log(x) + Sum_{k = 1..n} H(k)/k ). - _Peter Bala_, Feb 17 2022
%H A027459 Alexander Adamchuk, <a href="/A027459/b027459.txt">Table of n, a(n) for n = 1..30</a>
%H A027459 Jerry Metzger and Thomas Richards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Metzger/metz1.html">A Prisoner Problem Variation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
%H A027459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F A027459 Numerators of sequence a(1, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.
%F A027459 Numerators of (Wolstenholme(n, 1)^2 + Wolstenholme(n, 2))/(2*n)= ((gamma+Psi(n+1))^2 + Pi^2/6 - Psi(1, n+1))/(2*n), where Wolstenholme(n, m) = Sum_{i=1..n} 1/i^m. - _Vladeta Jovovic_, Aug 09 2002
%F A027459 a(n) = numerator(Sum_{k=1..n} ((Sum_{i=1..k} 1/i)/k)). - _Alexander Adamchuk_, Jan 02 2007
%e A027459 (a[ i,j ])^3 = MATRIX([[1, 0, 0, 0, 0], [7/8, 1/8, 0, 0, 0], [85/108, 19/108, 1/27, 0, 0], [415/576, 115/576, 37/576, 1/64, 0], [12019/18000, 3799/18000, 1489/18000, 61/2000, 1/125]]), n = 5.
%t A027459 Table[Numerator[Sum[Sum[1/i,{i,1,k}]/k,{k,1,n}]],{n,1,30}] (* _Alexander Adamchuk_, Jan 02 2007 *)
%t A027459 With[{nn=20},Accumulate[HarmonicNumber[Range[nn]]/Range[nn]]]//Numerator (* _Harvey P. Dale_, Feb 26 2023 *)
%o A027459 (Magma) [Numerator(&+[HarmonicNumber(k)/k:k in [1..n]]):n in [1..20]]; // _Marius A. Burtea_, Nov 05 2019
%Y A027459 Cf. A001008, A002805, A007406, A007407, A027447, A329108.
%K A027459 nonn
%O A027459 1,2
%A A027459 _Olivier Gérard_
%E A027459 Corrected by _Vladeta Jovovic_, Aug 09 2002