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

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

Author

Mats Granvik, Jan 06 2009

Keywords

• sign
• mult
• easy

Comments

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

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

Scheme

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

Formula

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

Extensions

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