# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a250310 Showing 1-1 of 1 %I A250310 #54 Jun 27 2022 19:06:06 %S A250310 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 A250310 98,100,110,112,118,124,128,130,134,136,140,142,146,148,152,158,164, %U A250310 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 A250310 Numbers whose squares are of the form x^2 + y^2 + 3 where x >= y >= 0 (repetitions omitted). %C A250310 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 A250310 Also because K^2-3 is the sum of two squares it must be congruent to 1 (mod 4). Consequently K is even. %C A250310 Numbers k such that k^2-3 is in A001481. - _Robert Israel_, Feb 05 2019 %H A250310 Robert Israel, Table of n, a(n) for n = 1..10000 %H A250310 Frank M. Jackson and Stalislav Takhaev, Heronian Triangles of Class K: Congruent Incircles Cevian Perspective, Forum Geom., 15 (2015) 5-12. %e A250310 a(4) = 10 as 10^2 - 3 = 9^2 + 4^2 and 10 is the 4th such occurrence. %p A250310 filter:= proc(n) local F; %p A250310 F:= ifactors(n^2-3)[2]; %p A250310 andmap(t -> t[1] mod 4 <> 3 or t[2]::even, F) %p A250310 end proc: %p A250310 select(filter, [seq(i,i=2..1000,2)]); # _Robert Israel_, Feb 05 2019 %t A250310 lst = {}; Do[If[IntegerQ[k=Sqrt[m^2+n^2+3]], AppendTo[lst, k]], {m, 0, 1000}, {n, 0, m}]; Union@lst %o A250310 (Python) %o A250310 from itertools import count, islice %o A250310 from sympy import factorint %o A250310 def A250310_gen(): # generator of terms %o A250310 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 A250310_list = list(islice(A250310_gen(),30)) # _Chai Wah Wu_, Jun 27 2022 %Y A250310 Cf. A034017, A210484. %K A250310 nonn %O A250310 1,1 %A A250310 _Frank M Jackson_ and Stalislav Takhaev, Jan 24 2015 %E A250310 Edited by _Robert Israel_, Feb 05 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE