%I #12 Jan 25 2023 22:23:13

%S 1,-1,-1,1,-1,1,-1,-2,0,1,-1,-1,-1,1,1,3,-1,0,-1,-1,1,1,-1,2,0,1,0,-1,

%T -1,-1,-1,-5,1,1,1,0,-1,1,1,2,-1,-1,-1,-1,0,1,-1,-3,0,0,1,-1,-1,0,1,2,

%U 1,1,-1,1,-1,1,0,8,1,-1,-1,-1,1,-1,-1,0,-1,1,0,-1,1,-1,-1,-3,0,1,-1,1,1,1,1,2,-1,0,1,-1,1,1,1,5,-1,0,0,0,-1,-1,-1,2,-1

%N Dirichlet inverse of A359832, where A359832(n) = 1 if the 2-adic valuation of n is either 0 or odd, otherwise 0.

%C Multiplicative because A359832 is.

%H Antti Karttunen, <a href="/A359833/b359833.txt">Table of n, a(n) for n = 1..65537</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A359832(n/d) * a(d).

%F Multiplicative with a(2^e) = (-1)^e*Fibonacci(e), and for p > 2, a(p^e) = -1 if e = 1 and 0 otherwise. - _Amiram Eldar_, Jan 24 2023

%t f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := (-1)^e*Fibonacci[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 24 2023 *)

%o (PARI) A359833(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k,1], ((-1)^f[k,2])*fibonacci(f[k,2]),-(1==f[k,2]))); }; \\ _Antti Karttunen_, Jan 25 2023

%Y Cf. A000045, A007814, A359832, A359834 (parity of terms).

%K sign,mult

%O 1,8

%A _Antti Karttunen_, Jan 24 2023