A033684 1 iff n is a square not divisible by 3.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

a(n)=1 iff n-1 is in the list A057780. - Jason Kimberley, Nov 13 2012

References

• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 105, Eq. (40).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65536
• Index entries for characteristic functions.

Crossrefs

Cf. A010052, A011655, A033683, A057780.

Programs

Maple

A033684 := proc(n)
        if issqr(n) then
                if n mod 3 = 0 then
                        0;
                else
                        1;
                end if;
        else
                0;
        end if;
end proc:
seq(A033684(n),n=0..80) ; # R. J. Mathar, Oct 07 2011

Mathematica

Table[If[IntegerQ[Sqrt[n]]&&Mod[n,3]!=0,1,0],{n,0,130}] (* Harvey P. Dale, Oct 19 2018 *)

PARI

A033684(n) = (issquare(n)&&(n%3)); \\ Antti Karttunen, Sep 13 2017

Formula

Essentially the series psi_3(z)=(1/2)(theta_3(z/9)-theta_3(z)).

a(n) * A000035(n) = A033683(n).

Multiplicative with a(p^e) = 1 if 2 divides e and p != 3, 0 otherwise. - Mitch Harris, Jun 09 2005

Dirichlet g.f.: zeta(2*s)*(1-3^(-2*s)). - R. J. Mathar, Mar 10 2011

a(n) = A010052(n)*A011655(n). - Antti Karttunen, Sep 13 2017

Sum_{k=1..n} a(k) ~ 2*sqrt(n)/3. - Amiram Eldar, Jan 14 2024

Extensions

Data-section extended up to a(121) by Antti Karttunen, Sep 13 2017