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A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).

%I A000499 M5193 N2257 #49 Jan 04 2025 06:18:45
%S A000499 0,1,27,184,875,2700,7546,17600,35721,72750,126445,223776,353717,
%T A000499 595448,843750,1349120,1827636,2808837,3600975,5306000,6667920,
%U A000499 9599172,11509982,16416000,19015625,26605670,30902310,41686848,46948825,64233000,70306760,94089216
%N A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).
%D A000499 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000499 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000499 Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H A000499 John Cerkan, <a href="/A000499/b000499.txt">Table of n, a(n) for n = 1..10000</a>
%H A000499 Jacques Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). - _Michel Marcus_, Feb 02 2014
%F A000499 a(n) = (n^3/24 - n^4/8)*sigma_1(n) + (n^3/12)*sigma_3(n). - _Ridouane Oudra_, Sep 15 2020
%F A000499 Sum_{k=1..n} a(k) ~ Pi^4 * n^7 / 7560. - _Vaclav Kotesovec_, Aug 08 2022
%e A000499 G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ...
%p A000499 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(3);
%t A000499 a[n_] := Sum[k^3*DivisorSigma[1, k]*DivisorSigma[1, n - k], {k, 1, n - 1}]; Array[a, 32] (* _Jean-François Alcover_, Feb 09 2016 *)
%o A000499 (PARI) a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ _Michel Marcus_, Feb 02 2014
%o A000499 (PARI) a(n) = my(f = factor(n)); ((n^3 - 3*n^4) * sigma(f) + 2*n^3 * sigma(f, 3)) / 24; \\ _Amiram Eldar_, Jan 04 2025
%Y A000499 Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.
%Y A000499 Cf. A000203 (sigma_1), A001158 (sigma_3).
%K A000499 nonn
%O A000499 1,3
%A A000499 _N. J. A. Sloane_
%E A000499 More terms and 0 prepended by _Michel Marcus_, Feb 02 2014