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%I #21 Dec 29 2022 06:31:10
%S 1,-1,1,0,1,-1,1,0,0,-1,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,0,-1,0,0,1,-1,1,
%T 0,1,-1,1,0,1,-1,1,0,1,-1,1,0,0,-1,1,0,0,0,1,0,1,0,1,0,1,-1,1,0,1,-1,
%U 0,0,1,-1,1,0,1,-1,1,0,1,-1,0,0,1,-1,1,0,0,-1,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,0,0,1,-1,1,0,1
%N Dirichlet inverse of A359154, where A359154 is multiplicative with a(p^e) = (-1)^(p*e).
%H Antti Karttunen, <a href="/A359155/b359155.txt">Table of n, a(n) for n = 1..100000</a>
%F Multiplicative with a(p) = (-1)^(1+p), and a(p^e) = 0 if e > 1.
%F a(n) = A008683(n) * A359154(n).
%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A359154(n/d) * a(d).
%F For all n >= 1, a(A003961(n)) = A008966(n).
%F Dirichlet g.f.: (zeta(s)/zeta(2*s))*((2^s-1)/(2^s+1)). - _Amiram Eldar_, Dec 29 2022
%t f[p_, e_] := If[e == 1, 1, 0]; f[2, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 29 2022 *)
%o (PARI) A359155(n) = { my(f = factor(n)); prod(k=1, #f~, (1==f[k,2])*((-1)^(1+f[k, 1]))); };
%o (PARI) A359155(n) = { my(f = factor(n)); moebius(n)*prod(k=1, #f~, (-1)^(f[k, 1]*f[k, 2])); };
%o (Python)
%o from functools import reduce
%o from operator import ixor
%o from sympy import factorint
%o def A359155(n): return 0 if max((f:=factorint(n)).values(),default=0) > 1 else -1 if reduce(ixor,(p&1^1 for p in f.keys()),0) else 1 # _Chai Wah Wu_, Dec 21 2022
%Y Cf. A001414, A003961, A008683, A008966, A359154.
%K sign,mult
%O 1
%A _Antti Karttunen_, Dec 19 2022
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