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A361983 a(n) = 1 + Sum_{k=2..n} (-1)^k * k^2 * a(floor(n/k)).

%I A361983 #21 May 10 2023 04:31:24
%S A361983 1,5,-4,28,3,-33,-82,174,174,74,-47,-335,-504,-700,-475,1573,1284,
%T A361983 1284,923,123,564,80,-449,-2753,-2753,-3429,-3429,-4997,-5838,-4938,
%U A361983 -5899,10485,11574,10418,11643,11643,10274,8830,10351,3951,2270,4034,2185,-1687,-1687,-3803
%N A361983 a(n) = 1 + Sum_{k=2..n} (-1)^k * k^2 * a(floor(n/k)).
%H A361983 Seiichi Manyama, <a href="/A361983/b361983.txt">Table of n, a(n) for n = 1..8191</a>
%F A361983 Sum_{k=1..n} (-1)^k * k^2 * a(floor(n/k)) = -1.
%F A361983 G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k^2 * (1 - x^k) * A(x^k).
%t A361983 f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := 2^(3*e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* _Amiram Eldar_, May 09 2023 *)
%o A361983 (Python)
%o A361983 from functools import lru_cache
%o A361983 @lru_cache(maxsize=None)
%o A361983 def A361983(n):
%o A361983     if n <= 1:
%o A361983         return 1
%o A361983     c, j = 1, 2
%o A361983     k1 = n//j
%o A361983     while k1 > 1:
%o A361983         j2 = n//k1 + 1
%o A361983         c += ((j2*(j2-1) if j2&1 else -j2*(j2-1))+(-j*(j-1) if j&1 else j*(j-1))>>1)*A361983(k1)
%o A361983         j, k1 = j2, n//j2
%o A361983     return c+((-n*(n+1) if n&1 else n*(n+1))+(-j*(j-1) if j&1 else j*(j-1))>>1) # _Chai Wah Wu_, Apr 02 2023
%Y A361983 Partial sums of A361987.
%Y A361983 Cf. A347030, A361982.
%Y A361983 Cf. A336276.
%K A361983 sign,look
%O A361983 1,2
%A A361983 _Seiichi Manyama_, Apr 02 2023