Nice Subsets

A *good number* is a positive integer which is not divisible by the cube (third power) of any prime. For example, $2$, $3$, $25 = 5 \cdot 5$, $63 = 7 \cdot 9$ and $2057 = 11 \cdot 11 \cdot 17$ are good numbers, but $8 = 2 \cdot 2 \cdot 2$, $9317 = 7 \cdot 11 \cdot 11 \cdot 11$ and $2401 = 7 \cdot 7 \cdot 7 \cdot 7$ are not good numbers. A pair of positive integers $(x, y)$ is *special* if $x \cdo