%I #85 May 10 2023 04:31:15

%S 1,-8,-35,-27,-152,91,-252,-252,-252,873,-458,-674,-2871,216,3591,

%T 3591,-1322,-1322,-8181,-9181,80,12059,-108,-108,-108,19665,19665,

%U 16921,-7468,-37843,-67634,-67634,-31697,12520,55395,55395,4742,66473,125792,125792,56871,-26478

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

%H Seiichi Manyama, <a href="/A360658/b360658.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{k=1..n} k^3 * a(floor(n/k)) = 0 for n > 1.

%F G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k).

%t 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 (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A360658(n):

%o if n <= 1:

%o return 1

%o c, j = 0, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A360658(k1)

%o j, k1 = j2, n//j2

%o return c-((n*(n+1))**2-((j-1)*j)**2>>2) # _Chai Wah Wu_, Apr 01 2023

%Y Partial sums of A359531.

%Y Cf. A092149, A359478, A360390.

%Y Cf. A336277.

%K sign

%O 1,2

%A _Seiichi Manyama_, Apr 01 2023