A239978 Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.

6, 30, 60, 72, 84, 126, 168, 180, 210, 210, 252, 252, 288, 330, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 504, 528, 528, 546, 624, 630, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 840, 864, 924, 924, 924, 924, 924, 936, 990, 990, 1008

Offset: 1

Author

Frank M Jackson, Mar 30 2014

Keywords

• nonn

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.

Examples

			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.
		

Links

• Paul Yiu, Heron triangles which cannot be decomposed into two integer right triangles, 2008.

Crossrefs

Cf. A083875, A224301, A227003, A227166.

Programs

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*)