A087049 Characteristic sequence for numbers n>=0 that are either squares or have a square > 1 as factor.

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

Offset: 0

Author

Wolfdieter Lang, Sep 08 2003

Keywords

• nonn
• easy

Comments

a(0)=1, a(1)=1, n>=2: a(n)=1 if isquarefree(n)=false else 0.

Except for a(0)=1 and a(1)=1 this is the bit-flipped unsigned Moebius sequence abs(A008683(n)), n>=2.

For n>=2: a(n)=1 iff n is from A013929 (not squarefree).

Examples

			a(4) = 1 because 4 is a square; a(8) = 1 because 8 = 2^2 * 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Index entries for characteristic functions.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A008683, A008966, A080733, A000290 (squares), A013929 (not squarefree), A229099.

Programs

Maple

1,1,seq(`if`(numtheory:-issqrfree(n),0,1),n=2..100); # Robert Israel, Nov 17 2017

Mathematica

Array[If[# <= 1, 1, 1 - Abs@ MoebiusMu@ #] &, 105, 0] (* Michael De Vlieger, Nov 17 2017 *)

PARI

A087049(n) = if(n<=1,1,1-abs(moebius(n))); \\ Antti Karttunen, Nov 17 2017

Formula

a(n) = 1 if n is a perfect square (A000290) or has some square > 1 as a factor, else 0.

a(0) = a(1) = 1; for n > 1, a(n) = 1 - A008966(n). - Antti Karttunen, Nov 17 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Jan 19 2024