Vtyjkyuk

A square number is a number $n$ with the property $S(n) :equiv (exists x) x^2 = n$.
A number $n neq 0$ is a square number iff it has an odd number of divisors.

A hypotenuse number is a number $n$ with the property $H(n) :equiv (exists x, y) x^2 + y^2 = n^2$.
A number is a hypotenuse number iff it has a prime factor of the form $4k+1$.

A Pythagorean triple is a triple $(a,b,c)$ of numbers with the property $P(a,b,c) :equiv a^2 + b^2 = c^2$.
A triple $(a,b,c)$ is Pythagorean iff

$$(*) (exists x,y,z) phi(x,y,z) wedge a = x(y^2 - z^2) wedge b = x(2yz) wedge c = x(y^2 + z^2)$$

with $phi(x,y,z) :equiv x,y,zgeq 0, y > z, y, z$ co-prime and not both odd.

I wonder if the following property $R(n)$ has been defined before in its own right and what its official name is.

$$R(n) :equiv (exists x,y,z) P(x,y,z) wedge x + y + z =n$$

A number $n$ is $R$ iff a regular $n$-gon with rigid sides can be transformed into a right triangle, possibly in different ways, thus being the perimeter of a right-angled triangle (thanks to Ronald Blaak for this formulation). ($R$ stands for "right-triangulizable".)

From $(*)$ it's obvious that $R(n)$ iff $(exists x,y,z) phi(x,y,z) wedge 2xy(y + z) = n$.

Is there another characterization of $R$ involving prime factors (instead of arbitrary divisors of which two have to be co-prime)?

The final characterisation of $R(n)$ :

$R(n)$ iff $(exists x,y,z) phi(x,y,z) land 2 x y (y+z)=n $

is correct but can be simplified. Using that $x,y,z > 0$ (excluding zero-length sides), the resulting values of $a,b,c$ should be positive in order result in a proper length and perimeter. This is only relevant for $a>0$ because it gives $y>z$.

Since the perimeter of these triangles is always even, we find
$fracn2=x y (y+z)$ has to be an integer, with $1 leq z < y < y+z$, and it follows that $2 leq y < y+z < 2 y$. In other words, the half-perimeter needs to have to at least two different factors $p,q$ such that $p < q < 2 p$.

This is also sufficient, since if $fracn2=p q r$ with $rgeq 1$, we can take $x=r$,$y=p$, and $z=q-p$ to find an example of a right-angled triangle with perimeter $n$.

This criterion is simpler(more efficient), in the sense that it does not assume any triplets $(x,y,z)$ in advance, but is acting directly on the perimeter $n$ and also constructs a valid $(a,b,c)$ triplet from the input $n$ and its factorisation when possible.

I am not aware of a special name for these numbers, but the sequence can be found in the encyclopaedia of integer sequences A010814.