A364268 a(n) = Sum_{k=1..n} k^2*sigma_2(k), where sigma_2 is A001157.

1, 21, 111, 447, 1097, 2897, 5347, 10787, 18158, 31158, 45920, 76160, 104890, 153890, 212390, 299686, 383496, 530916, 661598, 879998, 1100498, 1395738, 1676108, 2165708, 2572583, 3147183, 3744963, 4568163, 5276285, 6446285, 7370767, 8768527, 10097107

Offset

1,2

Links

Table of n, a(n) for n=1..33.

Wikipedia, Faulhaber's formula.

Formula

a(n) = Sum_{k=1..n} k^4 * A000330(floor(n/k)).

a(n) ~ (zeta(3)/5) * n^5. - Amiram Eldar, Oct 20 2023

Mathematica

Accumulate[Table[n^2*DivisorSigma[2, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)

Prog

(PARI) f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=2, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));
(Python)
def A364268(n): return sum(k**4*(m:=n//k)*(m+1)*((m<<1)+1)//6 for k in range(1, n+1)) # Chai Wah Wu, Oct 20 2023
(Python)
from math import isqrt
def A364268(n): return (((s:=isqrt(n))*(s+1)*(2*s+1))**2*(1-3*s*(s+1))//6 + sum((q:=n//k)*(q+1)*(2*q+1)*k**2*(5*k**2+3*q*(q+1)-1) for k in range(1, s+1)))//30 # Chai Wah Wu, Oct 21 2023

Crossrefs

Cf. A356125, A364269.

Cf. A000330, A001157.

Sequence in context: A157886 A039456 A182827 * A255285 A157265 A275916

Adjacent sequences: A364265 A364266 A364267 * A364269 A364270 A364271

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 20 2023

Status

approved