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A336277
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a(n) = Sum_{k=1..n} mu(k)*k^3.
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8
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1, -7, -34, -34, -159, 57, -286, -286, -286, 714, -617, -617, -2814, -70, 3305, 3305, -1608, -1608, -8467, -8467, 794, 11442, -725, -725, -725, 16851, 16851, 16851, -7538, -34538, -64329, -64329, -28392, 10912, 53787, 53787, 3134, 58006, 117325, 117325, 48404
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n) changes sign infinitely often.
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LINKS
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FORMULA
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G.f. A(x) satisfies x = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
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MATHEMATICA
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PROG
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(PARI) a(n) = sum(k=1, n, moebius(k)*k^3); \\ Michel Marcus, Jul 15 2020
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n <= 1:
return 1
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A336277(k1)
j, k1 = j2, n//j2
return c-((n*(n+1))**2-((j-1)*j)**2>>2) # Chai Wah Wu, Apr 04 2023
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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