A343785 a(n) is completely multiplicative with a(p^e) = (-1)^e if p == 2 (mod 3) and a(p^e) = 1 otherwise.

1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1

Offset

1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Peter Borwein and Michael Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv:0806.1694 [math.NT], 2008.

Michael J. Coons, Some aspects of analytic number theory: parity, transcendence, and multiplicative functions, Ph.D. Thesis, Department of Mathematics, Simon Fraser University, 2009.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, 2011, pages 571-614. a(n) = d(n) at equation 5.2 and multiplicative at equation 6.3.

Kevin Ryde, Iterations of the Terdragon Curve, see index "turn".

Formula

a(3*n) = a(n), a(3*n+1) = 1 and a(3*n+2) = -1.

G.f.: Sum_{k>=0} x^(3^k) / (1 + x^(3^k) + x^(2*3^k)).

Sum_{k>=1} a(k)/2^k = A343786.

From Peter Munn, Jun 10 2021: (Start)

a(n) = 1 if n is in A182828, otherwise a(n) = -1.

a(n) = A008836(A343430(n)). (End)

Mathematica

f[p_, e_] := If[Mod[p, 3] == 2, (-1)^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = my(r); until(r, [n, r]=divrem(n, 3)); -(-1)^r; \\ Kevin Ryde, Nov 09 2021
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A343785(n): return -1 if reduce(ixor, (e&1 if p%3==2 else 0 for p, e in factorint(n).items()), 0) else 1 # Chai Wah Wu, Dec 23 2022

Crossrefs

Cf. A209615, A182828, A343786.

A008836 composed with A343430.

Cf. A060236 (as 1,2), A080846 (as 0,1), A137893 (as 1,0).

Cf. A026179 (indices of -1's except n=1), A062756 (partial sums).

Sequence in context: A256175 A319116 A337004 * A359738 A360710 A269529

Adjacent sequences: A343782 A343783 A343784 * A343786 A343787 A343788

Keyword

sign,mult,easy

Author

Amiram Eldar, Apr 29 2021

Status

approved