

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

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(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[mx^2y^2]&&FP[x+2y2*Sqrt[mx^2y^2]], ex=Append[ex, m]; Goto[aa]], {x, 0, Sqrt[m]}, {y, 0, Sqrt[mx^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

ZhiWei Sun, Dec 15 2016


STATUS

approved



