A168181 Characteristic function of numbers that are not multiples of 8.

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

Offset

0,1

Comments

Multiplicative with a(p^e) = (if p=2 then A019590(e) else 1), p prime and e>0.

Period 8 Repeat: [0, 1, 1, 1, 1, 1, 1, 1]. - Wesley Ivan Hurt, Jun 21 2014

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

Index entries for characteristic functions

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Formula

a(n+8) = a(n);

a(n) = A000007(A010877(n));

a(A047592(n)) = 1; a(A008590(n)) = 0;

A033440(n) = Sum_{k=0..n} a(k)*(n-k).

Dirichlet g.f. (1-1/8^s)*zeta(s). - R. J. Mathar, Feb 19 2011

For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = sign(n mod 8). - Wesley Ivan Hurt, Jun 21 2014

a(n) = sign( 1 - floor(cos(Pi*n/4)) ). - Wesley Ivan Hurt, Jun 21 2014

Euler transform of length 8 sequence [ 1, 0, 0, 0, 0, 0, -1, 1]. - Michael Somos, Jun 24 2014

Moebius transform is length 8 sequence [ 1, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Jun 24 2014

G.f.: x * (1 - x^7) / ((1 - x) * (1 - x^8)). - Michael Somos, Jun 24 2014

a(n) = 1-A253513(n). - Antti Karttunen, Oct 08 2017

Example

G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + ...

Maple

with(numtheory); A168181:=n->signum(n mod 8); seq(A168181(n), n=0..100); # Wesley Ivan Hurt, Jun 21 2014

Mathematica

Table[Sign[Mod[n, 8]], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 21 2014 *)

Prog

(Magma) [Sign(n mod 8) : n in [0..100]]; // Wesley Ivan Hurt, Jun 21 2014
(PARI) a(n)=n%8 > 0 \\ Felix Fröhlich, Aug 11 2014
(Python)
def A168181(n): return int(bool(n&7)) # Chai Wah Wu, Jul 09 2022

Crossrefs

Cf. A168185, A145568, A168184, A168182, A109720, A097325, A011558, A166486, A011655, A000035, A010877, A244413, A253513.

Sequence in context: A377874 A374128 A304438 * A368916 A324732 A164980

Adjacent sequences: A168178 A168179 A168180 * A168182 A168183 A168184

Keyword

mult,nonn,easy

Author

Reinhard Zumkeller, Nov 30 2009

Status

approved