A279616 Numbers of the form x^2 + y^2 + z^2 with x + 2*y - 2*z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.

1, 3, 4, 5, 9, 10, 14, 16, 17, 18, 19, 20, 22, 24, 29, 33, 34, 35, 37, 41, 45, 48, 49, 50, 51, 52, 53, 58, 59, 61, 64, 65, 66, 68, 69, 70, 73, 74, 77, 78, 80, 82, 84, 88, 89, 90, 94, 97, 98, 99, 100, 104, 106, 107, 109, 113, 114, 116, 117, 121, 122, 125, 129, 130, 132, 133, 138, 139, 141, 144

Offset

1,2

Comments

Part (i) of the conjecture in A279612 implies that any positive integer can be written as the sum of a square and a term of the current sequence.

It seems that a(n)/n has the limit 2 as n tends to the infinity.

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

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
FP[n_]:=FP[n]=n>0&&IntegerQ[Log[4, n]];
ex={}; Do[Do[If[SQ[m-x^2-y^2]&&FP[x+2y-2*Sqrt[m-x^2-y^2]], ex=Append[ex, m]; Goto[aa]], {x, 0, Sqrt[m]}, {y, 0, Sqrt[m-x^2]}]; Label[aa]; Continue, {m, 1, 144}]

Crossrefs

Cf. A000079, A000290, A271518, A275656, A275675, A275676, A275738, A278560, A279612.

Sequence in context: A327257 A228941 A236211 * A050068 A250444 A327178

Adjacent sequences: A279613 A279614 A279615 * A279617 A279618 A279619

Keyword

nonn

Author

Zhi-Wei Sun, Dec 15 2016

Status

approved