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A154269 Dirichlet inverse of A019590; Fully multiplicative with a(2^e) = (-1)^e, a(p^e) = 0 for odd primes p.

%I A154269 #34 Jul 18 2025 08:34:37
%S A154269 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 A154269 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 A154269 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 A154269 Dirichlet inverse of A019590; Fully multiplicative with a(2^e) = (-1)^e, a(p^e) = 0 for odd primes p.
%C A154269 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 A154269 Mobius transform of A035263. - _R. J. Mathar_, Jul 14 2012
%H A154269 Mats Granvik (first 220 terms) & Antti Karttunen, <a href="/A154269/b154269.txt">Table of n, a(n) for n = 1..65536</a>
%F A154269 Abs(a(n)) = A036987(n-1) = A209229(n).
%F A154269 a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = 0^e if p>2. - _Michael Somos_, Jul 05 2009
%F A154269 G.f. A(x) satisfies x = A(x) + A(x^2).
%F A154269 Dirichlet g.f.: (1 + 2^(-s))^(-1). - _Michael Somos_, Jul 05 2009
%F A154269 a(1) = 1, after which: a(2n) = -a(n), a(2n+1) = 0. - _Antti Karttunen_, Jul 24 2017
%F A154269 a(n) = Sum_{d|n} A323239(d)*A008836(n/d). - _Ridouane Oudra_, Jul 15 2025
%e A154269 x - x^2 + x^4 - x^8 + x^16 - x^32 + x^64 - x^128 + x^256 - x^512 + ...
%p A154269 a:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
%p A154269 seq(a(n), n=1..95);  # _Alois P. Heinz_, Feb 18 2024
%t A154269 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 A154269 (PARI) {a(n) = if( n < 2, n == 1, - a(n / 2))} /* _Michael Somos_, Jul 05 2009 */
%o A154269 (Scheme) (define (A154269 n) (cond ((= 1 n) 1) ((even? n) (* -1 (A154269 (/ n 2)))) (else 0))) ;; _Antti Karttunen_, Jul 24 2017
%Y A154269 Cf. A209229 (gives the absolute values).
%Y A154269 Cf. A035263, A154271, A154282, A323239, A008836.
%K A154269 sign,mult
%O A154269 1,1
%A A154269 _Mats Granvik_, Jan 06 2009
%E A154269 Alternative description added to the name by _Antti Karttunen_, Jul 24 2017