Perfect Right-angled Triangles
Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$. The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and $28$.
Moreover it is a primitive right angled triangle as $\gcd(a,b)=1$ and $\gcd(b,c)=1$.
Also $c$ is a perfect square.
We will call a right angled triangle perfect if
-it is a primitive right angled triangle
-its hypotenuse is a perfect square.
We will call a right angled triangle super-perfect if
-it is a perfect right angled triangle and
-its area is a multiple of the perfect numbers $6$ and $28$.
How many perfect right-angled triangles with $c \le 10^{16}$ exist that are not super-perfect?