A350108 a(n) = Sum_{k=1..n} k * floor(n/k)^3.

1, 10, 32, 87, 153, 309, 443, 722, 1005, 1443, 1785, 2605, 3087, 3951, 4875, 6154, 6988, 8809, 9855, 12057, 13853, 16001, 17543, 21347, 23478, 26484, 29440, 33696, 36162, 41994, 44816, 50351, 54755, 59909, 64577, 73524, 77558, 84002, 90142, 100072, 105034

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{k=1..n} k * Sum_{d|k} (d^3 - (d - 1)^3)/d.

G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 - x^k)^2.

From Vaclav Kotesovec, Aug 03 2022: (Start)

a(n) = A024916(n) + 3*A143128(n) - 3*A143127(n).

a(n) ~ Pi^2*n^3/6 - 3*n^2*log(n)/2. (End)

Mathematica

a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, k*(n\k)^3);
(PARI) a(n) = sum(k=1, n, k*sumdiv(k, d, (d^3-(d-1)^3)/d));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))
(Python)
from math import isqrt
def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1, s+1))>>1 # Chai Wah Wu, Oct 31 2023

Crossrefs

Column 3 of A350106.

Cf. A024916, A143128, A143127, A318742.

Sequence in context: A202804 A155192 A229720 * A024933 A198646 A356277

Adjacent sequences: A350105 A350106 A350107 * A350109 A350110 A350111

Keyword

nonn

Author

Seiichi Manyama, Dec 14 2021

Status

approved