A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

1, 41, 311, 1655, 4905, 15705, 32855, 76375, 142714, 272714, 435096, 797976, 1171466, 1857466, 2734966, 4131702, 5556472, 8210032, 10692990, 15060990, 19691490, 26186770, 32635280, 44385680, 54557555, 69497155, 85637215, 108686815, 129222353, 164322353

Offset

1,2

Links

Table of n, a(n) for n=1..30.

Wikipedia, Faulhaber's formula.

Formula

a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).

a(n) ~ (zeta(3)/6) * n^6. - Amiram Eldar, Oct 20 2023

Mathematica

Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* 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=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));
(Python)
def A364269(n): return sum(k*(k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1, n+1)) # Chai Wah Wu, Oct 20 2023
(Python)
from math import isqrt
def A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1, s+1)))//12 # Chai Wah Wu, Oct 21 2023

Crossrefs

Cf. A356125, A364268.

Cf. A000537, A001157.

Sequence in context: A074281 A090833 A201043 * A002646 A175110 A096170

Adjacent sequences: A364266 A364267 A364268 * A364270 A364271 A364272

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 20 2023

Status

approved