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

Author

Reinhard Zumkeller, Nov 30 2009

Keywords

• mult
• nonn
• easy

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

Examples

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

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).

Crossrefs

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

Programs

Magma

[Sign(n mod 8) : n in [0..100]]; // Wesley Ivan Hurt, Jun 21 2014

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 *)

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

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