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

%I A033684 #42 Jan 14 2024 03:12:08
%S A033684 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,
%T A033684 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,
%U A033684 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
%N A033684 1 iff n is a square not divisible by 3.
%C A033684 a(n)=1 iff n-1 is in the list A057780. - _Jason Kimberley_, Nov 13 2012
%D A033684 J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 105, Eq. (40).
%H A033684 Antti Karttunen, <a href="/A033684/b033684.txt">Table of n, a(n) for n = 0..65536</a>
%H A033684 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A033684 Essentially the series psi_3(z)=(1/2)(theta_3(z/9)-theta_3(z)).
%F A033684 a(n) * A000035(n) = A033683(n).
%F A033684 Multiplicative with a(p^e) = 1 if 2 divides e and p != 3, 0 otherwise. - _Mitch Harris_, Jun 09 2005
%F A033684 Dirichlet g.f.: zeta(2*s)*(1-3^(-2*s)). - _R. J. Mathar_, Mar 10 2011
%F A033684 a(n) = A010052(n)*A011655(n). - _Antti Karttunen_, Sep 13 2017
%F A033684 Sum_{k=1..n} a(k) ~ 2*sqrt(n)/3. - _Amiram Eldar_, Jan 14 2024
%p A033684 A033684 := proc(n)
%p A033684         if issqr(n) then
%p A033684                 if n mod 3 = 0 then
%p A033684                         0;
%p A033684                 else
%p A033684                         1;
%p A033684                 end if;
%p A033684         else
%p A033684                 0;
%p A033684         end if;
%p A033684 end proc:
%p A033684 seq(A033684(n),n=0..80) ; # _R. J. Mathar_, Oct 07 2011
%t A033684 Table[If[IntegerQ[Sqrt[n]]&&Mod[n,3]!=0,1,0],{n,0,130}] (* _Harvey P. Dale_, Oct 19 2018 *)
%o A033684 (PARI) A033684(n) = (issquare(n)&&(n%3)); \\ _Antti Karttunen_, Sep 13 2017
%Y A033684 Cf. A010052, A011655, A033683, A057780.
%K A033684 nonn,easy,mult
%O A033684 0,1
%A A033684 _N. J. A. Sloane_
%E A033684 Data-section extended up to a(121) by _Antti Karttunen_, Sep 13 2017