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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 28 15:04 EDT 2022. Contains 354115 sequences. (Running on oeis4.)