%I #30 Feb 18 2024 10:28:51
%S 1,-1,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Dirichlet inverse of A019590; Fully multiplicative with a(2^e) = (-1)^e, a(p^e) = 0 for odd primes p.
%C Equals +1 if n is an even power of 2 (2^0, 2^2, 2^4,...), -1 if n is an odd power of 2 (2^1, 2^3, 2^5,..) and zero anywhere else.
%C Mobius transform of A035263. - _R. J. Mathar_, Jul 14 2012
%H Mats Granvik (first 220 terms) & Antti Karttunen, <a href="/A154269/b154269.txt">Table of n, a(n) for n = 1..65536</a>
%F Abs(a(n)) = A036987(n-1) = A209229(n).
%F a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = 0^e if p>2. - _Michael Somos_, Jul 05 2009
%F G.f. A(x) satisfies x = A(x) + A(x^2).
%F Dirichlet g.f.: (1 + 2^(-s))^(-1). - _Michael Somos_, Jul 05 2009
%F a(1) = 1, after which: a(2n) = -a(n), a(2n+1) = 0. - _Antti Karttunen_, Jul 24 2017
%e x - x^2 + x^4 - x^8 + x^16 - x^32 + x^64 - x^128 + x^256 - x^512 + ...
%p a:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
%p seq(a(n), n=1..95); # _Alois P. Heinz_, Feb 18 2024
%t nn = 95;a = PadRight[{1, 1}, nn, 0];Inverse[Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]][[All, 1]] (* _Mats Granvik_, Jul 24 2017 *)
%o (PARI) {a(n) = if( n < 2, n == 1, - a(n / 2))} /* _Michael Somos_, Jul 05 2009 */
%o (Scheme) (define (A154269 n) (cond ((= 1 n) 1) ((even? n) (* -1 (A154269 (/ n 2)))) (else 0))) ;; _Antti Karttunen_, Jul 24 2017
%Y Cf. A209229 (gives the absolute values).
%Y Cf. A035263, A154271, A154282.
%K sign,mult
%O 1,1
%A _Mats Granvik_, Jan 06 2009
%E Alternative description added to the name by _Antti Karttunen_, Jul 24 2017