Let $C(n)$ be a function that counts composites $\leq n$ , $P(n)$ be a function that counts primes $\leq n$, and $s$ be a perfect square.
How many squares $s$ have the property $C(s)/P(s)$ is an integer?
For example, with $s=3^2=9$, we have $C(9) = \{4,6,8,9\} = 4$ and $P(9) = {2,3,5,7} = 4$, so $C/P = 4/4 = 1$.
Also, $s=11^2=121$ and $s=19^2=361$ have the property that $P(s)$ divides $C(s)$ evenly, but I find no other $s \leq 30000^2$.
Is there a number theory argument for $9$, $121$, and $361$ being the only possible squares?