Integers with prescribed divisor count

Question

1. Given an integer $n\in\Bbb N$ is there always an integer with $n^2$ divisors?

2. How to find such integers?

Answer 1

For $m=p_1^{a_1}p_2^{a_2}\cdots p_s^{a_s}$ we have $$ \tau(m)=(a_1+1)(a_2+1)\cdots(a_s+1)=\prod_{j=1}^s(a_j+1), $$ where $\tau(m)$ denotes the number of positive divisors of $m$. So for $m=2^{n-1}3^{n-1}$ we have $$ \tau(m)=n^2. $$ But indeed, $\tau(2^{n-1})=n$, so every $n$ appears as the number of divisors of some $m$.