

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.


5



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
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OFFSET

1,3


COMMENTS

This is motivated by the author's 135Conjecture 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 GaussLegendre theorem on sums of three squares, no term has the form 4^k*(8m+7) with k and m nonnegative integers.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare 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[mx^2y^2]&&SQ[x+3y+5*Sqrt[mx^2y^2]], n=n+1; Print[n, " ", m]; Goto[aa]], {x, 0, Sqrt[m]}, {y, 0, Sqrt[mx^2]}]; Label[aa]; Continue, {m, 0, 146}]


CROSSREFS

Cf. A000290, A271518, A273294, A273302.
Sequence in context: A253317 A277971 A080288 * A217682 A169868 A191159
Adjacent sequences: A278557 A278558 A278559 * A278561 A278562 A278563


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 23 2016


STATUS

approved



