4+ Classical combinatorics (other).

A natural number $b$ is called 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.
Lemma. A number $b$ is lucky if and only if each prime number enters the prime factorization of $b$ with one of the following exponents: $0,1,2,3,4,6,8$.

Proof. Let's call a non-negative integer $k$ happy if there does not exist an integer $m$ such that $2 m < k < 2 m + 2$. If $k > 4$, then choose the largest number $m$ such that $2 m < k$. Since $k > 4$, and $2,5 m > 2 m + 2 = 2(m + 1) > k$ by the choice of $m$, then $k$ is unhappy.

Suppose the number $b$ is unlucky, that is, $a^{5}$ is divisible by $b^{2}$, but $a^{2}$ is not divisible by $b$ for some $a$. Then some prime $p$ enters the factorization of $a^{2}$ in a smaller degree than in the factorization of $b$. Let $p$ enter the factorizations of $a$ and $b$ in degrees $m$ and $k$ respectively; then $2 m < 2 k$ and $5 m \geq 2 k$. Therefore, $2 k - 2 m \geq 2$ and $5 m - 2 k \geq 0$, which implies $2 k - 2 m \leq 2(5 m - 2 k)$, or $3(2 k - 2 m) \leq 2(5 m - 2 k)$. Hence, $2 k - 2 m \leq 2$, and $2 k - 2 m \geq 2$, which means $2 k - 2 m = 2$, or $k - m = 1$. Therefore, $k \geq 5$ and $k$ is unhappy.

Since $2^{5} > 2010$ and $5^{5} > 2010$, each unlucky number less than 2010 belongs to one of the following non-intersecting classes:

1) numbers of the form $2^{5} q$, where $q$ is odd and $q \leq 61\left(2^{5} \cdot 61 < 2010 < 2^{5} \cdot 63\right)$;
2) numbers of the form $2^{7} q$, where $q$ is odd and $q \leq 15\left(2^{7} \cdot 15 < 2010 < 2^{7} \cdot 17\right)$;
3) numbers of the form $2^{9} q$, where $q = 1$ or $3\left(2^{9} \cdot 3 < 2010 < 2^{9} \cdot 5\right)$;
4) the number $2^{10}$;
5) numbers of the form $3^{5} q$, where $q$ is not divisible by 3 and $q \leq 8\left(3^{5} \cdot 8 < 2010 < 3^{5} \cdot 10\right)$.

Thus, the total number of unlucky numbers less than 2010 is $31 + 8 + 2 + 1 + 6 = 48$, and the number of lucky numbers is

$2009 - 48 = 1961$