%I #19 Jan 24 2023 16:44:33

%S 1,-1,-1,0,0,1,-1,0,1,0,0,0,0,1,0,0,0,-1,0,0,1,0,0,0,0,0,-1,0,0,0,-1,

%T 0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1

%N Dirichlet inverse of A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.

%H Antti Karttunen, <a href="/A359579/b359579.txt">Table of n, a(n) for n = 1..107163</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A336923(n/d) * a(d).

%F Multiplicative with a(2^e) = -1 if e = 1, and 0 if e > 1, and for primes p > 2, a(p^e) = (-A036987(p))^e. - Corrected by _Amiram Eldar_ and _Antti Karttunen_, Jan 24 2023

%F For all n >= 1, abs(a(A056652(n))) = abs(a(2*A056652(n))) = 1.

%F For all n >= 1, abs(a(A219174(n))) = 1 if A219174(n) is not a multiple of 4.

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

%o (PARI)

%o A209229(n) = (n && !bitand(n,n-1));

%o A359579(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1], -(1==f[k,2]), (-A209229(1+f[k,1]))^f[k,2])); };

%Y Cf. A036987, A056652, A219174, A336923.

%Y Cf. also A359578.

%K sign,mult

%O 1

%A _Antti Karttunen_, Jan 08 2023