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A250310 Numbers whose squares are of the form x^2 + y^2 + 3 where x >= y >= 0 (repetitions omitted). 2
2, 4, 8, 10, 14, 20, 22, 26, 32, 34, 40, 44, 46, 52, 56, 58, 64, 68, 74, 80, 86, 88, 92, 94, 98, 100, 110, 112, 118, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 158, 164, 172, 178, 184, 190, 194, 202, 206, 208, 212, 218, 220, 230, 238, 242, 244, 250, 254, 256, 266, 268, 274, 278, 290, 296, 298 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There exists a K-class of Heronian triangles such that the sum of the tangents of their half angles is a constant K > 1, iff K^2-3 is the sum of two squares. E.g., for K = 2 (x=1, y=0) we generate the class of integer Soddyian triangles (see A034017, A210484). For K = 4 (x=2, y=3) the class generated is Heronian triangles with the ratio of r_i : r_o : r = 1 : 3 : 6 where r is their inradius and r_i, r_o are the radii of their inner and outer Soddy circles.

Also because K^2-3 is the sum of two squares it must be congruent to 1 (mod 4). Consequently K is even.

Numbers k such that k^2-3 is in A001481. - Robert Israel, Feb 05 2019

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Frank M. Jackson, Stalislav Takhaev, Heronian Triangles of Class K: Congruent Incircles Cevian Perspective, Forum Geom., 15 (2015) 5-12.

EXAMPLE

a(4) = 10 as 10^2 - 3 = 9^2 + 4^2 and 10 is the 4th such occurrence.

MAPLE

filter:= proc(n) local F;

  F:= ifactors(n^2-3)[2];

  andmap(t -> t[1] mod 4 <> 3 or t[2]::even, F)

end proc:

select(filter, [seq(i, i=2..1000, 2)]); # Robert Israel, Feb 05 2019

MATHEMATICA

lst = {}; Do[If[IntegerQ[k=Sqrt[m^2+n^2+3]], AppendTo[lst, k]], {m, 0, 1000}, {n, 0, m}]; Union@lst

CROSSREFS

Cf. A034017, A210484.

Sequence in context: A317626 A292550 A024895 * A288447 A087915 A088967

Adjacent sequences:  A250307 A250308 A250309 * A250311 A250312 A250313

KEYWORD

nonn

AUTHOR

Frank M Jackson and Stalislav Takhaev, Jan 24 2015

EXTENSIONS

Edited by Robert Israel, Feb 05 2019

STATUS

approved

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Last modified October 25 16:41 EDT 2021. Contains 348255 sequences. (Running on oeis4.)