We call a natural number $b$ lucky if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of lucky natural numbers less than $2010$.
Source: Russian Regional Olympiad 2010 10.4
Tags: number theory, divisible
We call a natural number $b$ lucky if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of lucky natural numbers less than $2010$.