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A334147 Numbers which can be written uniquely as x^4 + y*(2y+1) + z*(3z+1), where x,y,z are integers with x>=0.

%I A334147 #13 Apr 17 2020 11:27:53
%S A334147 0,9,42,57,127,218,243,272,412,467,554,555,571,724,909,1292,1385,1448,
%T A334147 1557,1604,1897,2062,2410,3025,3507,4328,5907,8182,9018,14654,18628,
%U A334147 25479,25713,76322,80488,152177,1277405
%N A334147 Numbers which can be written uniquely as x^4 + y*(2y+1) + z*(3z+1), where x,y,z are integers with x>=0.
%C A334147 The sequence consists of those n with A334138(n) = 1.
%C A334147 Conjecture: The sequence has no terms greater than a(37) = 1277405.
%C A334147 We have noted that A334138(n) > 1 for all 1277405 < n <= 5*10^6.
%e A334147 a(10) = 467 with 467 = 0^4 + 15*(2*15+1) + (-1)*(3*(-1)+1).
%e A334147 a(25) = 3507 with 3507 = 6^4 + 33*(2*33+1) + 0*(3*0+1).
%e A334147 a(36) = 152177 with 152177 = 9^4 + (-266)*(2*(-266)+1) + 38*(3*38+1).
%e A334147 a(37) = 1277405 with 1277405 = 22^4 + (-655)*(2*(-655)+1) + (-249)*(3*(-249)+1).
%t A334147 QQ[n_]:=QQ[n]=IntegerQ[Sqrt[12n+1]];
%t A334147 m=0;Do[r=0;Do[If[QQ[n-x^4-y(2y+1)],r=r+1;If[r>1,Goto[aa]]],{x,0,n^(1/4)},{y,-Floor[(Sqrt[8(n-x^4)+1]+1)/4],(Sqrt[8(n-x^4)+1]-1)/4}];If[r==1,m=m+1;Print[m," ",n]];Label[aa],{n,0,152177}]
%Y A334147 Cf. A000217, A001318, A270566, A334138.
%K A334147 nonn
%O A334147 1,2
%A A334147 _Zhi-Wei Sun_, Apr 16 2020