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A336279 a(n) = Sum_{k=1..n} mu(k)*k^5.

%I A336279 #28 Apr 04 2023 22:09:19
%S A336279 1,-31,-274,-274,-3399,4377,-12430,-12430,-12430,87570,-73481,-73481,
%T A336279 -444774,93050,852425,852425,-567432,-567432,-3043531,-3043531,
%U A336279 1040570,6194202,-242141,-242141,-242141,11639235,11639235,11639235,-8871914,-33171914,-61801065
%N A336279 a(n) = Sum_{k=1..n} mu(k)*k^5.
%C A336279 Conjecture: a(n) changes sign infinitely often.
%H A336279 Seiichi Manyama, <a href="/A336279/b336279.txt">Table of n, a(n) for n = 1..10000</a>
%F A336279 From _Seiichi Manyama_, Apr 03 2023: (Start)
%F A336279 G.f. A(x) satisfies x = Sum_{k>=1} k^5 * (1 - x^k) * A(x^k).
%F A336279 Sum_{k=1..n} k^5 * a(floor(n/k)) = 1. (End)
%t A336279 Array[Sum[MoebiusMu[k]*k^5, {k, #}] &, 32] (* _Michael De Vlieger_, Jul 15 2020 *)
%o A336279 (PARI) a(n) = sum(k=1, n, moebius(k)*k^5); \\ _Michel Marcus_, Jul 15 2020
%o A336279 (Python)
%o A336279 from functools import lru_cache
%o A336279 @lru_cache(maxsize=None)
%o A336279 def A336279(n):
%o A336279     if n <= 1:
%o A336279         return 1
%o A336279     c, j = 1, 2
%o A336279     k1 = n//j
%o A336279     while k1 > 1:
%o A336279         j2 = n//k1 + 1
%o A336279         c -= (j2**2*(j2**2*(j2*(2*j2 - 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12*A336279(k1)
%o A336279         j, k1 = j2, n//j2
%o A336279     return c-(n**2*(n**2*(n*(2*n + 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12 # _Chai Wah Wu_, Apr 04 2023
%Y A336279 Cf. A002321, A068340, A336276, A336277, A336278.
%Y A336279 Cf. A008683, A055615, A070891, A344429, A344430.
%K A336279 easy,sign
%O A336279 1,2
%A A336279 _Donald S. McDonald_, Jul 15 2020