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a(1) = 1, a(2) = -5; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.
5

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

%S 1,-5,-9,4,-25,45,-49,0,0,125,-121,-36,-169,245,225,0,-289,0,-361,

%T -100,441,605,-529,0,0,845,0,-196,-841,-1125,-961,0,1089,1445,1225,0,

%U -1369,1805,1521,0,-1681,-2205,-1849,-484,0,2645,-2209,0,0,0,2601,-676,-2809,0,3025,0,3249,4205,-3481,900,-3721,4805,0

%N a(1) = 1, a(2) = -5; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.

%H Seiichi Manyama, <a href="/A359485/b359485.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) is multiplicative with a(2)= -5, a(4)= 4, a(2^e)= 0 if e>2. a(p)= -p^2, a(p^e)= 0 if e>1, p>2.

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

%t f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := Switch[e, 1, -5, 2, 4, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 10 2023 *)

%Y Partial sums give A360390.

%Y Cf. A092673, A359484, A359531.

%Y Cf. A334657.

%K sign,mult

%O 1,2

%A _Seiichi Manyama_, Apr 01 2023