OFFSET
1,1
COMMENTS
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.
See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1008 are captured.
LINKS
EXAMPLE
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.
MATHEMATICA
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*)
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Mar 30 2014
STATUS
approved