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

1,3

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000290, A271518, A273294, A273302.

Sequence in context: A253317 A277971 A080288 * A217682 A169868 A363088

Adjacent sequences: A278557 A278558 A278559 * A278561 A278562 A278563

Keyword

nonn

Author

Zhi-Wei Sun, Nov 23 2016

Status

approved