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A318742
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a(n) = Sum_{k=1..n} floor(n/k)^3.
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11
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1, 9, 29, 74, 136, 254, 382, 596, 833, 1173, 1505, 2057, 2527, 3209, 3921, 4856, 5674, 6928, 7956, 9474, 10882, 12608, 14128, 16506, 18369, 20797, 23141, 26129, 28567, 32259, 35051, 38963, 42483, 46675, 50435, 55904, 59902, 65156, 70092, 76460
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) ~ zeta(3) * n^3.
G.f.: (1/(1 - x)) * Sum_{k>=1} (3*k*(k - 1) + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019
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MATHEMATICA
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Table[Sum[Floor[n/k]^3, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 3*DivisorSigma[1, k] + 3*DivisorSigma[2, k], {k, 1, 40}]]
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PROG
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(PARI) a(n) = sum(k=1, n, (n\k)^3); \\ Michel Marcus, Sep 03 2018
(Magma) [&+[Floor(n/k)^3:k in [1..n]]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019
(Python)
from math import isqrt
def A318742(n): return -(s:=isqrt(n))**4 + sum((q:=n//k)*(3*k*(k-1)+q**2+1) for k in range(1, s+1)) # Chai Wah Wu, Oct 21 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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