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A364269
a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.
3
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
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
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Oct 20 2023
STATUS
approved