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
Links
- Wikipedia, Faulhaber's formula.
Programs
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Mathematica
Accumulate[Table[n^2*DivisorSigma[2, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *) -
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));
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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
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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
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