Problem 4. Given a positive integer $n$, denote by $\tau(n)$ the number of positive divisors of $n$, and by $\sigma(n)$ the sum of all positive divisors of $n$. Find all positive integers $n$ satisfying

$$
\sigma(n)=\tau(n) \cdot\lceil\sqrt{n}\rceil
$$

(Here, $\lceil x\rceil$ denotes the smallest integer not less than $x$.)

(Michael Reitmeir)