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%I #16 Oct 05 2023 14:34:53
%S 0,1,51,472,2963,10764,36538,95936,222561,502638,974245,1850784,
%T 3234269,5826680,8857926,15093248,21945012,35369541,48358119,74448464,
%U 98697648,148971972,187495262,276509952,336495665,488970662,590163894,823791168,966018241,1358404776
%N a(n) = Sum_{k=1..n-1} k^4*sigma(k)*sigma(n-k).
%C This was formerly A001477.
%H Colin Barker, <a href="/A259692/b259692.txt">Table of n, a(n) for n = 1..1000</a>
%H J. Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F From _Ridouane Oudra_, Sep 09 2023: (Start)
%F a(n) = (n^4/24 - n^5/10)*sigma_1(n) + (5*n^4/84)*sigma_3(n) - (691/635040)*sigma_5(n) - (13/127008)*sigma_11(n) + (691/2520)*A279889(n).
%F a(n) = (n^4/24 - n^5/10)*sigma_1(n) - (691/1512000 - 5*n^4/84)*sigma_3(n) - (691/756000)*sigma_7(n) + (13/72000)*sigma_11(n) - (691/3150)*A279964(n).
%F a(n) = (-691/1596672 + n^4/24 - n^5/10)*sigma_1(n) + (5*n^4/84)*sigma_3(n) - (691/145152 - 691*n/120960)*sigma_9(n) - (65/38016)*sigma_11(n) + (691/6048)*f(n), where f(n) = Sum_{k=1..n-1} sigma_1(k)*sigma_9(n-k). (End)
%p 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(4);
%t a[n_]:=Sum[k^4*DivisorSigma[1,k]*DivisorSigma[1,n-k],{k,1,n-1}]; Table[a[n],{n,1,30}] (* _Robert P. P. McKone_, Sep 09 2023 *)
%o (PARI) a(n) = sum(k=1, n-1, k^4*sigma(k)*sigma(n-k)) \\ _Colin Barker_, Jul 16 2015
%Y Cf. A000385, A000441, A000477, A000499, A259693, A259694, A259695, A259696.
%Y Cf. A279889, A279964.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Jul 03 2015
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