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A275168 Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

%I A275168 #18 Feb 28 2018 16:58:15
%S A275168 6,18,23,41,42,59,78,86,96,114,115,123,142,187,195,205,213,214,240,
%T A275168 261,262,266,303,322,329,330,383,423,478,501,510,581,610,618,642,682,
%U A275168 690,698,761,774,807,865,870,906,959,963,990,1206,1222,1230,1234,1302,1312,1314,1320,1346,1411,1697,1706,1781
%N A275168 Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.
%C A275168 Conjecture: The sequence has totally 150 terms as listed in the b-file the largest of which is 182842. Thus any integer n > 182842 can be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.
%C A275168 We note that the sequence has no term greater than 182842 and not exceeding 10^6.
%C A275168 See also A275169 for a similar conjecture.
%C A275168 It is known that for any positive integers a,b,c there are infinitely many positive integers not of the form a*x^2 + b*y^2 + c*z^2 with x,y,z nonnegative integers.
%H A275168 Zhi-Wei Sun, <a href="/A275168/b275168.txt">Table of n, a(n) for n = 1..150</a>
%e A275168 a(1) = 6 since 1 = 0^3 + 3*0^2 + 1^2, 2 = 1^3 + 3*0^2 + 1^2, 3 = 0^3 + 3*1^2 + 0^2, 4 = 0^3 + 3*1^2 + 1^2, 5 = 1^3 + 3*1^2 + 1^2, but 6 cannot be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.
%t A275168 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A275168 n=0;Do[Do[If[SQ[m-x^3-3*y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/3]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,1800}]
%Y A275168 Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275169.
%K A275168 nonn
%O A275168 1,1
%A A275168 _Zhi-Wei Sun_, Jul 18 2016