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A360658 a(1) = 1; a(n) = -Sum_{k=2..n} k^3 * a(floor(n/k)).

%I A360658 #85 May 10 2023 04:31:15
%S A360658 1,-8,-35,-27,-152,91,-252,-252,-252,873,-458,-674,-2871,216,3591,
%T A360658 3591,-1322,-1322,-8181,-9181,80,12059,-108,-108,-108,19665,19665,
%U A360658 16921,-7468,-37843,-67634,-67634,-31697,12520,55395,55395,4742,66473,125792,125792,56871,-26478
%N A360658 a(1) = 1; a(n) = -Sum_{k=2..n} k^3 * a(floor(n/k)).
%H A360658 Seiichi Manyama, <a href="/A360658/b360658.txt">Table of n, a(n) for n = 1..10000</a>
%F A360658 Sum_{k=1..n} k^3 * a(floor(n/k)) = 0 for n > 1.
%F A360658 G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k).
%t A360658 f[p_, e_] := If[e == 1, -p^3, 0]; f[2, e_] := Switch[e, 1, -9, 2, 8, _, 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* _Amiram Eldar_, May 10 2023 *)
%o A360658 (Python)
%o A360658 from functools import lru_cache
%o A360658 @lru_cache(maxsize=None)
%o A360658 def A360658(n):
%o A360658     if n <= 1:
%o A360658         return 1
%o A360658     c, j = 0, 2
%o A360658     k1 = n//j
%o A360658     while k1 > 1:
%o A360658         j2 = n//k1 + 1
%o A360658         c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A360658(k1)
%o A360658         j, k1 = j2, n//j2
%o A360658     return c-((n*(n+1))**2-((j-1)*j)**2>>2) # _Chai Wah Wu_, Apr 01 2023
%Y A360658 Partial sums of A359531.
%Y A360658 Cf. A092149, A359478, A360390.
%Y A360658 Cf. A336277.
%K A360658 sign
%O A360658 1,2
%A A360658 _Seiichi Manyama_, Apr 01 2023