A227166 Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

72, 126, 168, 252, 252, 288, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 528, 528, 624, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 864, 924, 924, 924, 924, 936, 990, 1008, 1092, 1092, 1188, 1200, 1218, 1248, 1260, 1260, 1320, 1320, 1320

Offset: 1

Author

Frank M Jackson, Jul 03 2013

Keywords

• nonn

Comments

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

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

Examples

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

Links

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

Crossrefs

Cf. A083875, A224301, A227003, A239978.

Programs

Mathematica

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 *)

Extensions

Name clarified by Frank M Jackson, Mar 17 2014