A064603 Partial sums of A001158: Sum_{j=1..n} sigma_3(j).

1, 10, 38, 111, 237, 489, 833, 1418, 2175, 3309, 4641, 6685, 8883, 11979, 15507, 20188, 25102, 31915, 38775, 47973, 57605, 69593, 81761, 98141, 113892, 133674, 154114, 179226, 203616, 235368, 265160, 302609, 339905, 384131, 427475, 482736

Offset: 1

Author

Labos Elemer, Sep 24 2001

Keywords

• nonn

Comments

In general, Sum_{k=1..n} sigma_m(k) = Sum_{k=1..n} k^m * floor(n/k). - Daniel Suteu, Nov 08 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A001158, A064606.

Cf. A064602, A064604, A248076.

Programs

Mathematica

Accumulate@ Array[DivisorSigma[3, #] &, 36] (* Michael De Vlieger, Nov 03 2017 *)

PARI

a(n) = sum(j=1, n, sigma(j, 3)); \\ Michel Marcus, Nov 04 2017

PARI

a(n) = sum(k=1, n, k^3 * (n\k)); \\ Daniel Suteu, Nov 08 2018

Python

from math import isqrt
def A064603(n): return (-(s:=isqrt(n))**3*(s+1)**2 + sum((q:=n//k)*(4*k**3+q*(q*(q+2)+1)) for k in range(1,s+1)))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = a(n-1) + A001158(n) = Sum_{j=1..n} sigma_3(j), where sigma_3(j) = A001158(j).

G.f.: (1/(1 - x))*Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 23 2017

a(n) ~ Pi^4 * n^4 / 360. - Vaclav Kotesovec, Sep 02 2018

a(n) = Sum_{k=1..n} ((1/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018

a(n) = Sum_{k=1..n} k^3 * floor(n/k). - Daniel Suteu, Nov 08 2018