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 A000477 M4973 N2135 #56 Jan 04 2025 06:19:00
%S A000477 0,1,15,76,275,720,1666,3440,6129,11250,17545,28896,41405,65072,85950,
%T A000477 128960,162996,238545,286995,404600,482160,662112,756470,1042560,
%U A000477 1150625,1549730,1732590,2257920,2443105,3250800,3421160,4452096,4791600,6039522,6296500
%N A000477 a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).
%D A000477 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000477 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000477 Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H A000477 John Cerkan, <a href="/A000477/b000477.txt">Table of n, a(n) for n = 1..10000</a>
%H A000477 Jacques Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F A000477 a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k). - _Sean A. Irvine_, Nov 14 2010
%F A000477 G.f.: x*f(x)*g'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k) and g(x) = Sum_{k>=1} k^2*x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, May 02 2018
%F A000477 a(n) = (n^2/24 - n^3/6)*sigma_1(n) + (n^2/8)*sigma_3(n). - _Ridouane Oudra_, Sep 15 2020
%F A000477 Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 4320. - _Vaclav Kotesovec_, May 09 2022
%e A000477 G.f. = x^2 + 15*x^3 + 76*x^4 + 275*x^5 + 720*x^6 + 1666*x^7 + 3440*x^8 + ...
%p A000477 with(numtheory): S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1); f:=e->[seq(S(n,e),n=1..30)]; f(2); # _N. J. A. Sloane_, Jul 03 2015
%t A000477 a[n_] := Sum[k^2 DivisorSigma[1, k] DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 35] (* _Jean-François Alcover_, Feb 08 2016 *)
%o A000477 (PARI) a(n) = sum(k=1, n-1, k^2*sigma(k)*sigma(n-k)); \\ _Michel Marcus_, Feb 02 2014
%o A000477 (PARI) a(n) = my(f = factor(n)); ((n^2 - 4*n^3) * sigma(f) + 3*n^2 * sigma(f, 3)) / 24; \\ _Amiram Eldar_, Jan 04 2025
%Y A000477 Cf. A000385, A000441, A000499, A259692, A259693, A259694, A259695, A259696.
%Y A000477 Cf. A000203 (sigma_1), A001158 (sigma_3).
%K A000477 nonn
%O A000477 1,3
%A A000477 _N. J. A. Sloane_
%E A000477 More terms from _Sean A. Irvine_, Nov 14 2010
%E A000477 a(1)=0 prepended by _Michel Marcus_, Feb 02 2014