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A360390 a(1) = 1; a(n) = -Sum_{k=2..n} k^2 * a(floor(n/k)).

%I A360390 #59 May 10 2023 04:31:11
%S A360390 1,-4,-13,-9,-34,11,-38,-38,-38,87,-34,-70,-239,6,231,231,-58,-58,
%T A360390 -419,-519,-78,527,-2,-2,-2,843,843,647,-194,-1319,-2280,-2280,-1191,
%U A360390 254,1479,1479,110,1915,3436,3436,1755,-450,-2299,-2783,-2783,-138,-2347,-2347,-2347,-2347,254,-422
%N A360390 a(1) = 1; a(n) = -Sum_{k=2..n} k^2 * a(floor(n/k)).
%H A360390 Seiichi Manyama, <a href="/A360390/b360390.txt">Table of n, a(n) for n = 1..10000</a>
%F A360390 Sum_{k=1..n} k^2 * a(floor(n/k)) = 0 for n > 1.
%F A360390 G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^2 * (1 - x^k) * A(x^k).
%t A360390 f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := Switch[e, 1, -5, 2, 4, _, 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* _Amiram Eldar_, May 10 2023 *)
%o A360390 (Python)
%o A360390 from functools import lru_cache
%o A360390 @lru_cache(maxsize=None)
%o A360390 def A360390(n):
%o A360390     if n <= 1:
%o A360390         return 1
%o A360390     c, j = 0, 2
%o A360390     k1 = n//j
%o A360390     while k1 > 1:
%o A360390         j2 = n//k1 + 1
%o A360390         c -= (j2*(j2-1)*((j2<<1)-1)-j*(j-1)*((j<<1)-1))//6*A360390(k1)
%o A360390         j, k1 = j2, n//j2
%o A360390     return c-(n*(n+1)*((n<<1)+1)-j*(j-1)*((j<<1)-1))//6 # _Chai Wah Wu_, Apr 01 2023
%Y A360390 Partial sums of A359485.
%Y A360390 Cf. A092149, A359478, A360658.
%Y A360390 Cf. A336276.
%K A360390 sign
%O A360390 1,2
%A A360390 _Seiichi Manyama_, Apr 01 2023