A259693 a(n) = Sum_{k=1..n-1} k^5*sigma(k)*sigma(n-k).

0, 1, 99, 1264, 10475, 44820, 185626, 546560, 1454841, 3640950, 7868245, 16042176, 31040789, 59796968, 97525350, 177090560, 276689076, 467100189, 681356055, 1096023200, 1533162960, 2426544252, 3205401854, 4885539840, 6250705625, 9431254430, 11831779350

Offset: 1

Author

N. J. A. Sloane, Jul 03 2015

Keywords

• nonn

Comments

This was formerly A001478.

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, A259692, 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(5);

Mathematica

S[n_, e_] := Sum[k^e * DivisorSigma[1, k] * DivisorSigma[1, n - k], {k, 1, n - 1}]
f[e_] := Table[S[n, e], {n, 1, 27}];f[5] (* James C. McMahon, Dec 19 2023 *)

PARI

a(n) = sum(k=1, n-1, k^5*sigma(k)*sigma(n-k)) \\ Colin Barker, Jul 16 2015

Formula

From Ridouane Oudra, Dec 08 2023: (Start)

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).

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).

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)