A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

1, 5, 12, 23, 36, 54, 74, 97, 125, 156, 186, 226, 268, 306, 354, 409, 458, 515, 574, 636, 710, 778, 838, 922, 1013, 1086, 1168, 1264, 1350, 1452, 1556, 1651, 1762, 1864, 1966, 2105, 2234, 2332, 2448, 2594, 2726, 2864, 3004, 3132, 3294, 3444, 3564, 3736, 3917, 4067

Offset

1,2

Links

Table of n, a(n) for n=1..50.

Formula

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

a(n) = 1 + Sum_{k=1..n-1} (sigma(k) + sigma(k+1)) for n > 0.

a(n) ~ (Pi*n)^2/6. - Vaclav Kotesovec, Jun 24 2021

Mathematica

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

Prog

(Magma) [&+[Floor(n/k)*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020
(PARI) a(n) = sum(k=1, n, my(q=n/k); floor(q) * ceil(q)); \\ Michel Marcus, Feb 17 2020
(Python)
from math import isqrt
from sympy import divisor_sigma
def A332569(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1, s+1))-divisor_sigma(n) # Chai Wah Wu, Oct 23 2023

Crossrefs

Row sums of A085383.

Cf. A000203, A006218, A006590, A024916, A049697, A222548, A332490.

Sequence in context: A298791 A054307 A172295 * A126573 A341209 A000327

Adjacent sequences: A332566 A332567 A332568 * A332570 A332571 A332572

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 16 2020

Status

approved