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

1,1

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.

Links

Table of n, a(n) for n=1..48.

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

Example

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

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 *)

Crossrefs

Cf. A083875, A224301, A227003, A239978.

Sequence in context: A291805 A266345 A143741 * A250772 A345541 A345794

Adjacent sequences: A227163 A227164 A227165 * A227167 A227168 A227169

Keyword

nonn

Author

Frank M Jackson, Jul 03 2013

Extensions

Name clarified by Frank M Jackson, Mar 17 2014

Status

approved