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.

0, 9, 42, 57, 127, 218, 243, 272, 412, 467, 554, 555, 571, 724, 909, 1292, 1385, 1448, 1557, 1604, 1897, 2062, 2410, 3025, 3507, 4328, 5907, 8182, 9018, 14654, 18628, 25479, 25713, 76322, 80488, 152177, 1277405

Offset: 1

Author

Zhi-Wei Sun, Apr 16 2020

Keywords

• nonn

Comments

The sequence consists of those n with A334138(n) = 1.

Conjecture: The sequence has no terms greater than a(37) = 1277405.

We have noted that A334138(n) > 1 for all 1277405 < n <= 5*10^6.

Examples

			a(10) = 467 with 467 = 0^4 + 15*(2*15+1) + (-1)*(3*(-1)+1).
a(25) = 3507 with 3507 = 6^4 + 33*(2*33+1) + 0*(3*0+1).
a(36) = 152177 with 152177 = 9^4 + (-266)*(2*(-266)+1) + 38*(3*38+1).
a(37) = 1277405 with 1277405 = 22^4 + (-655)*(2*(-655)+1) + (-249)*(3*(-249)+1).
		

Crossrefs

Cf. A000217, A001318, A270566, A334138.

Programs

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[12n+1]];
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}]