Eccoci.

The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to $12$.

Find the smallest integer that can be the length of a cathetus of exactly $47547$ different integer sided right-angled triangles.