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For Heronian triangles t with the perimeter p and area a, there are five distinct triangles with the area equal to the perimeter:

If a=p, count(t) = 5, if a=2p, count(t)=18.

For the area being an integer multiple of the perimeter, what is the sequence given by count(t), ie starting 5,18,...

source: Numberphile video 12min 43sec

https://www.youtube.com/watch?v=UIjeCKPHbso

Jamie M
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    The sequence is this: OEIS A007237 – Jaap Scherphuis Jan 29 '20 at 09:54
  • It might or might not be relevant, but, with the semiperimeter being $s=p/2$, the triangle's inradius is $r=a/s$; that is, the set of all such triangles with a specified area to perimeter ratio is the set of all such triangles with a specified inradius. Which might not necessarily be an integer. – Rosie F Feb 07 '20 at 18:30

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