login
a(n) = 1 + Sum_{k=2..n} (-1)^k * k^2 * a(floor(n/k)).
5

%I #21 May 10 2023 04:31:24

%S 1,5,-4,28,3,-33,-82,174,174,74,-47,-335,-504,-700,-475,1573,1284,

%T 1284,923,123,564,80,-449,-2753,-2753,-3429,-3429,-4997,-5838,-4938,

%U -5899,10485,11574,10418,11643,11643,10274,8830,10351,3951,2270,4034,2185,-1687,-1687,-3803

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

%H Seiichi Manyama, <a href="/A361983/b361983.txt">Table of n, a(n) for n = 1..8191</a>

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

%F G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k^2 * (1 - x^k) * A(x^k).

%t 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 (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A361983(n):

%o if n <= 1:

%o return 1

%o c, j = 1, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

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

%o 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 Partial sums of A361987.

%Y Cf. A347030, A361982.

%Y Cf. A336276.

%K sign,look

%O 1,2

%A _Seiichi Manyama_, Apr 02 2023