For all positive integer $n$, let $P(n)$ be the product of all (positive) divisors of $n$.
It is well known that $P(n)=n^{d(n)/2}$ where $d(n)$ denotes the number of divisors of $n$.
It is also known that $P$ is an injective map (see for example Here).
Now my question : what is the range of $P$ ? In other words, what are the positive integers $q$ which can be written $P(n)$ for some $n$ ?
If $n = \prod_{i=1}^k\, p_i^{e_i}$, where the $p_i>1$ are distinct primes and the $e_i\geq 1$ are their exponents, then
\begin{align} P(n) &= {\left(\prod_{i=1}^k\, p_i^{e_i}\right)}^{^{\left(\displaystyle\frac12\,\prod_{i=1}^k\,(1+e_i)\right)}}. \end{align}
Hence, the range of $P$ is
$$ \left\{m \in \mathbb N\,\middle|\, (\exists k\geq 1)\, (\exists \, \text{ exponents }\, e_1,e_2, \dots, e_k\geq 1\, \text{ and distinct primes }\, p_1,p_2,\dots,p_k)\,\\ m = {\left(\prod_{i=1}^k\, p_i^{e_i}\right)}^{\left(\displaystyle\frac12\,\prod_{i=1}^k\,(1+e_i)\right)} \right\}.$$
In other words, for each choice of integer $k\geq 1$, integer exponents $e_1,\dots,e_k\geq 1$ and distinct primes $p_1,\dots,p_k>0$ there is a corresponding $m$ in the range of $P$, and vice versa.