A259692 a(n) = Sum_{k=1..n-1} k^4*sigma(k)*sigma(n-k).

0, 1, 51, 472, 2963, 10764, 36538, 95936, 222561, 502638, 974245, 1850784, 3234269, 5826680, 8857926, 15093248, 21945012, 35369541, 48358119, 74448464, 98697648, 148971972, 187495262, 276509952, 336495665, 488970662, 590163894, 823791168, 966018241, 1358404776

Offset: 1

Author

N. J. A. Sloane, Jul 03 2015

Keywords

• nonn

Comments

This was formerly A001477.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Crossrefs

Cf. A000385, A000441, A000477, A000499, A259693, A259694, A259695, A259696.

Cf. A279889, A279964.

Programs

Maple

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);

Mathematica

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 *)

PARI

a(n) = sum(k=1, n-1, k^4*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

Formula

From Ridouane Oudra, Sep 09 2023: (Start)

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).

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).

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)