A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

1, 8, 22, 57, 91, 185, 247, 402, 545, 775, 917, 1379, 1573, 1995, 2455, 3106, 3428, 4377, 4775, 5909, 6753, 7727, 8301, 10331, 11230, 12564, 13904, 15990, 16888, 19908, 20930, 23597, 25545, 27767, 29827, 34468, 35910, 38660, 41328, 46318, 48080, 53644, 55578

Offset: 1

Author

Seiichi Manyama, Dec 15 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A064602, A143128, A222548, A350107, A350124, A350125, A350128.

Programs

Mathematica

Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)

PARI

a(n) = sum(k=1, n, k^2*(n\k)^2);

PARI

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));

PARI

a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

Python

from math import isqrt
def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (2*d - 1)/d^2 = Sum_{k=1..n} 2 * k * sigma(k) - sigma_2(k) = 2 * A143128(n) - A064602(n).

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

a(n) ~ n^3 * (Pi^2/9 - zeta(3)/3). - Vaclav Kotesovec, Dec 16 2021