A332490 a(n) = Sum_{k=1..n} k * ceiling(n/k).

1, 4, 10, 18, 30, 42, 61, 77, 101, 124, 153, 177, 218, 246, 285, 325, 373, 409, 467, 507, 570, 624, 683, 731, 816, 873, 942, 1010, 1095, 1155, 1258, 1322, 1418, 1500, 1589, 1673, 1801, 1877, 1976, 2072, 2203, 2287, 2426, 2514, 2643, 2767, 2886, 2982, 3155, 3262

Offset: 1

Author

Ilya Gutkovskiy, Feb 16 2020

Keywords

• nonn

Crossrefs

Cf. A000203, A000217, A006590, A024916.

Programs

Magma

[&+[k*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020

Mathematica

Table[Sum[k Ceiling[n/k], {k, 1, n}], {n, 1, 50}]
Table[n (n + 1)/2 + Sum[DivisorSigma[1, k], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[x/(1 - x)^3 + (x/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sum(k=1, n, k*ceil(n/k)); \\ Michel Marcus, Feb 17 2020

Python

from math import isqrt
def A332490(n): return n*(n+1)-(s:=isqrt(n-1))**2*(s+1)+sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 22 2023

Formula

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

a(n) = n*(n + 1)/2 + Sum_{k=1..n-1} sigma(k).

a(n) ~ (6 + Pi^2)*n^2/12. - Vaclav Kotesovec, Mar 10 2020