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

1,1

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

Table of n, a(n) for n=1..51.

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

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

Cf. A083875, A224301, A227003, A227166.

Sequence in context: A014203 A044083 A377992 * A024406 A024365 A057229

Adjacent sequences: A239975 A239976 A239977 * A239979 A239980 A239981

Keyword

nonn

Author

Frank M Jackson, Mar 30 2014

Status

approved