%I #24 Apr 12 2018 05:05:56
%S 1,5,29,65
%N Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.
%C The sequence has no term in the interval [66, 10^6].
%C Conjecture 1: The sequence only has the four terms 1, 5, 29 and 65.
%C Conjecture 2: For any integer n > 1 which is neither 17 nor a power of 2, if n = u^2 + 2*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with z nonzero.
%C Conjecture 3: For any positive integer n not of the form 4^k*m (k = 0,1,2,... and m = 1, 7, 13), if n = u^2 + 3*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with y nonzero.
%H Zhi-Wei Sun, <a href="/A301858/b301858.txt">Table of n, a(n) for n = 1..4</a>
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[If[QQ[m]==False,Goto[aa]];Do[If[SQ[m-2x^2-y^2],Goto[aa]],{x,1,Sqrt[m/2]},{y,0,Sqrt[(m-2x^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,1000}];Print[tab]
%Y Cf. A000290, A000549, A001481, A002479, A301471.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 27 2018
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