4. find all natural numbers $n>1$ such that the number of positive divisors of $n$ is equal to the third smallest positive divisor of $n$.

## Solution:
Let $n=p_{1}^{a_{1}} \cdots p_{k}^{a_{k}}$ with $p_{1}p_{1}^{2}, a_{1}=a_{2}=p_{1}-1 \geq 2$. The numbers we are looking for are therefore exactly those of the form

$$
\begin{array}{ll}
n=p^{p^{2}-1} & \text { for } p \text { prime, } \\
n=(p q)^{p-1} & \text { for prime numbers } p, q \text { with } q>p^{2}, p \geq 3
\end{array}
$$