Question 10
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)
$$
\sigma(n) = \tau(n) \cdot \lceil\sqrt{n}\rceil.
$$
(Here, $\lceil x \rceil$ denotes the smallest integer not less than $x$.)
(Michael Reitmeir)