A227166 - OEIS

%I #25 Apr 01 2014 03:30:29

%S 72,126,168,252,252,288,336,336,396,396,420,420,420,420,456,462,528,

%T 528,624,714,720,720,756,792,798,840,840,840,840,864,924,924,924,924,

%U 936,990,1008,1092,1092,1188,1200,1218,1248,1260,1260,1320,1320,1320

%N Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

%C An indecomposable integer Heronian triangle that is not Pythagorean cannot be decomposed into two separate Pythagorean triangles because it has no integer altitudes.

%C See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1320 are captured.

%H Paul Yiu, <a href="http://math.fau.edu/yiu/Southern080216.pdf">Heron triangles which cannot be decomposed into two integer right triangles</a>, 2008

%e a(2) = 126 as this is the second smallest area of an indecomposable non-Pythagorean primitive Heronian triangle. The triple is (5,51,52).

%t nn=1320; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s]&&GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]] && !IntegerQ[2Sqrt[area2]/a] && !IntegerQ[2Sqrt[area2]/b] && !IntegerQ[2Sqrt[area2]/c], AppendTo[lst, Sqrt[area2]]]], {a, 3, nn}, {b, a}, {c, b}]; Sort@Select[lst, #<=nn &] (* using _T. D. Noe_'s program A083875 *)

%Y Cf. A083875, A224301, A227003, A239978.

%K nonn

%O 1,1

%A _Frank M Jackson_, Jul 03 2013

%E Name clarified by _Frank M Jackson_, Mar 17 2014