A259693 - OEIS

%I #22 Jan 20 2024 09:19:35

%S 0,1,99,1264,10475,44820,185626,546560,1454841,3640950,7868245,

%T 16042176,31040789,59796968,97525350,177090560,276689076,467100189,

%U 681356055,1096023200,1533162960,2426544252,3205401854,4885539840,6250705625,9431254430,11831779350

%N a(n) = Sum_{k=1..n-1} k^5*sigma(k)*sigma(n-k).

%C This was formerly A001478.

%H Colin Barker, <a href="/A259693/b259693.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_, Dec 08 2023: (Start)

%F a(n) = (n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112)*sigma_3(n) - (691*n/254016)*sigma_5(n) - (65*n/254016)*sigma_11(n) + (691*n/1008)*A279889(n).

%F a(n) = (n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112 - 691*n/604800)*sigma_3(n) - (691*n/302400)*sigma_7(n) + (13*n/28800)*sigma_11(n) - (691*n/1260)*A279964(n).

%F a(n) = (-3455*n/3193344 + n^5/24 - n^6/12)*sigma_1(n) + (5*n^5/112)*sigma_3(n) + (-3455*n/290304 + 691*n^2/48384)*sigma_9(n) - (325*n/76032)*sigma_11(n) + (3455*n/12096)*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(5);

%t S[n_, e_] := Sum[k^e * DivisorSigma[1, k] * DivisorSigma[1, n - k], {k, 1, n - 1}]

%t f[e_] := Table[S[n, e], {n, 1, 27}];f[5] (* _James C. McMahon_, Dec 19 2023 *)

%o (PARI) a(n) = sum(k=1, n-1, k^5*sigma(k)*sigma(n-k)) \\ _Colin Barker_, Jul 16 2015

%Y Cf. A000385, A000441, A000477, A000499, A259692, A259694, A259695, A259696.

%Y Cf. A279889, A279964.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Jul 03 2015