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

6, 18, 23, 41, 42, 59, 78, 86, 96, 114, 115, 123, 142, 187, 195, 205, 213, 214, 240, 261, 262, 266, 303, 322, 329, 330, 383, 423, 478, 501, 510, 581, 610, 618, 642, 682, 690, 698, 761, 774, 807, 865, 870, 906, 959, 963, 990, 1206, 1222, 1230, 1234, 1302, 1312, 1314, 1320, 1346, 1411, 1697, 1706, 1781

Offset: 1

Author

Zhi-Wei Sun, Jul 18 2016

Keywords

• nonn

Comments

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.

We note that the sequence has no term greater than 182842 and not exceeding 10^6.

See also A275169 for a similar conjecture.

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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275169.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
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}]