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%I #12 Jan 21 2023 02:14:09
%S 6,30,60,72,84,126,168,180,210,210,252,252,288,330,336,336,396,396,
%T 420,420,420,420,456,462,504,528,528,546,624,630,714,720,720,756,792,
%U 798,840,840,840,840,840,864,924,924,924,924,924,936,990,990,1008
%N Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.
%C An indecomposable Heronian triangle is a Heronian triangle that cannot be split into two Pythagorean triangles. In other words, it has no integer altitude that is not a side of the triangle. Note that all primitive Pythagorean triangles are indecomposable.
%C See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1008 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(5) = 84 as this is the fifth ordered area of an indecomposable primitive Heronian triangle. The triple is (7,24,25) and it is Pythagorean.
%t nn=1008; 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])||(c^2+b^2==a^2)), 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, A227166.
%K nonn
%O 1,1
%A _Frank M Jackson_, Mar 30 2014
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