A278560 Numbers of the form x^2 + y^2 + z^2 with x + 3*y + 5*z a square, where x, y and z are nonnegative integers.

0, 1, 2, 3, 8, 9, 10, 13, 14, 16, 17, 19, 21, 25, 26, 29, 30, 32, 37, 38, 40, 41, 42, 46, 48, 49, 50, 51, 54, 58, 59, 65, 66, 69, 70, 72, 73, 74, 77, 78, 81, 83, 85, 89, 90, 97, 98, 101, 102, 104, 105, 106, 109, 114, 117, 118, 120, 122, 125, 128, 129, 130, 131, 134, 136, 138, 139, 144, 145, 146

Offset: 1

Author

Zhi-Wei Sun, Nov 23 2016

Keywords

• nonn

Comments

This is motivated by the author's 1-3-5-Conjecture which states that any nonnegative integer can be expressed as the sum of a square and a term of the current sequence.

Clearly, any term times a fourth power is also a term of this sequence. By the Gauss-Legendre theorem on sums of three squares, no term has the form 4^k*(8m+7) with k and m nonnegative integers.

Examples

			a(4) = 3 since 3 = 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.
a(5) = 8 since 8 = 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Crossrefs

Cf. A000290, A271518, A273294, A273302.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[m-x^2-y^2]&&SQ[x+3y+5*Sqrt[m-x^2-y^2]],n=n+1;Print[n," ",m];Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]}];Label[aa];Continue,{m,0,146}]