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A336278
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a(n) = Sum_{k=1..n} mu(k)*k^4.
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7
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1, -15, -96, -96, -721, 575, -1826, -1826, -1826, 8174, -6467, -6467, -35028, 3388, 54013, 54013, -29508, -29508, -159829, -159829, 34652, 268908, -10933, -10933, -10933, 446043, 446043, 446043, -261238, -1071238, -1994759, -1994759, -808838, 527498, 2028123
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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^4 * (1 - x^k) * A(x^k).
Sum_{k=1..n} k^4 * a(floor(n/k)) = 1. (End)
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MATHEMATICA
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Accumulate[Table[MoebiusMu[x]x^4, {x, 40}]] (* Harvey P. Dale, Jan 14 2021 *)
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PROG
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(PARI) a(n) = sum(k=1, n, moebius(k)*k^4); \\ 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**2*(j2*(6*j2 - 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30*A336278(k1)
j, k1 = j2, n//j2
return c-(n*(n**2*(n*(6*n + 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30 # 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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