A154282 -id:A154282 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Mats Granvik, Jan 06 2009

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.

Mobius transform of A035263. - R. J. Mathar, Jul 14 2012

			x - x^2 + x^4 - x^8 + x^16 - x^32 + x^64 - x^128 + x^256 - x^512 + ...

Mats Granvik (first 220 terms) & Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A209229 (gives the absolute values).

Cf. A035263, A154271, A154282, A323239, A008836.

a:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
seq(a(n), n=1..95);  # Alois P. Heinz, Feb 18 2024

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 *)

{a(n) = if( n < 2, n == 1, - a(n / 2))} /* Michael Somos, Jul 05 2009 */

(define (A154269 n) (cond ((= 1 n) 1) ((even? n) (* -1 (A154269 (/ n 2)))) (else 0))) ;; Antti Karttunen, Jul 24 2017

Abs(a(n)) = A036987(n-1) = A209229(n).
a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = 0^e if p>2. - Michael Somos, Jul 05 2009
G.f. A(x) satisfies x = A(x) + A(x^2).
Dirichlet g.f.: (1 + 2^(-s))^(-1). - Michael Somos, Jul 05 2009
a(1) = 1, after which: a(2n) = -a(n), a(2n+1) = 0. - Antti Karttunen, Jul 24 2017
a(n) = Sum_{d|n} A323239(d)*A008836(n/d). - Ridouane Oudra, Jul 15 2025

Alternative description added to the name by Antti Karttunen, Jul 24 2017

A154271 Dirichlet inverse of A154272; Fully multiplicative with a(3^e) = (-1)^e, a(p^e) = 0 for primes p <> 3.

Original entry on oeis.org

1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 06 2009

Abs(A154272) is a Fredholm-Rueppel-like sequence.

Sequence equals +1 if n is an even power of 3 (3^0, 3^2, 3^4,...), equals -1 if n is an odd power of 3 (3^1, 3^3, 3^5, 3^7,...) and zero elsewhere. - Comment edited by R. J. Mathar, Jun 24 2013

Antti Karttunen, Table of n, a(n) for n = 1..59049 (terms 1..220 from Mats Granvik)

Cf. A154272, A154269, A014578 (Möbius inverse), A154282, A225569.
Cf. A225569 (gives the absolute values when interpreted as the characteristic function of powers of 3, i.e., with starting offset 1 instead of 0).
Cf. A359377, A008836.

nn = 95;a = PadRight[{1, 0, 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 *)

A154271(n) = { my(k=valuation(n,3)); if((3^k)==n,(-1)^k,0); }; \\ Antti Karttunen, Jul 24 2017

(define (A154271 n) (cond ((= 1 n) 1) ((zero? (modulo n 3)) (* -1 (A154271 (/ n 3)))) (else 0))) ;; Antti Karttunen, Jul 24 2017

Fully multiplicative with a(3) = -1, a(p) = 0 for primes p <> 3. - Antti Karttunen, Jul 24 2017
From Amiram Eldar, Nov 03 2023: (Start)
abs(a(n)) = A225569(n-1).
Dirichlet g.f.: 1/(1+3^(-s)). (End)
a(n) = Sum_{d|n} A359377(d)*A008836(n/d). - Ridouane Oudra, Jul 15 2025

Alternative description added to the name by Antti Karttunen, Jul 24 2017

A154281 1,0,0,1 followed by 0,0,0...

Original entry on oeis.org

1, 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, 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, 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, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 06 2009

Dirichlet inverse of this sequence is A154282.

Cf. A154272, A154271, A154269.

PadRight[{1,0,0,1},120,0] (* Harvey P. Dale, Aug 06 2012 *)

a(n)={n==1 || n==4} \\ Andrew Howroyd, Aug 05 2018

Multiplicative with a(2^2) = 1, a(2^e) = 0 for e<>2, a(p^e) = 0 for odd prime p. - Andrew Howroyd, Aug 05 2018

cp's OEIS Frontend

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

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A154271 Dirichlet inverse of A154272; Fully multiplicative with a(3^e) = (-1)^e, a(p^e) = 0 for primes p <> 3.

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A154281 1,0,0,1 followed by 0,0,0...

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