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%I #54 Jun 27 2022 19:06:06
%S 2,4,8,10,14,20,22,26,32,34,40,44,46,52,56,58,64,68,74,80,86,88,92,94,
%T 98,100,110,112,118,124,128,130,134,136,140,142,146,148,152,158,164,
%U 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
%N Numbers whose squares are of the form x^2 + y^2 + 3 where x >= y >= 0 (repetitions omitted).
%C 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.
%C Also because K^2-3 is the sum of two squares it must be congruent to 1 (mod 4). Consequently K is even.
%C Numbers k such that k^2-3 is in A001481. - _Robert Israel_, Feb 05 2019
%H Robert Israel, <a href="/A250310/b250310.txt">Table of n, a(n) for n = 1..10000</a>
%H Frank M. Jackson and Stalislav Takhaev, <a href="http://forumgeom.fau.edu/FG2015volume15/FG201502index.html">Heronian Triangles of Class K: Congruent Incircles Cevian Perspective</a>, Forum Geom., 15 (2015) 5-12.
%e a(4) = 10 as 10^2 - 3 = 9^2 + 4^2 and 10 is the 4th such occurrence.
%p filter:= proc(n) local F;
%p F:= ifactors(n^2-3)[2];
%p andmap(t -> t[1] mod 4 <> 3 or t[2]::even, F)
%p end proc:
%p select(filter, [seq(i,i=2..1000,2)]); # _Robert Israel_, Feb 05 2019
%t lst = {}; Do[If[IntegerQ[k=Sqrt[m^2+n^2+3]], AppendTo[lst, k]], {m, 0, 1000}, {n, 0, m}]; Union@lst
%o (Python)
%o from itertools import count, islice
%o from sympy import factorint
%o def A250310_gen(): # generator of terms
%o return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n**2-3).items()),count(2))
%o A250310_list = list(islice(A250310_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%Y Cf. A034017, A210484.
%K nonn
%O 1,1
%A _Frank M Jackson_ and Stalislav Takhaev, Jan 24 2015
%E Edited by _Robert Israel_, Feb 05 2019
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