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

1,1

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)

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)

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 *)

Prog

(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

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).

Sequence in context: A015549 A015829 A016334 * A225569 A087032 A236677

Adjacent sequences: A154268 A154269 A154270 * A154272 A154273 A154274

Keyword

sign,mult,easy

Author

Mats Granvik, Jan 06 2009

Extensions

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

Status

approved