Smallest Numbers with 240 Divisors

Theorem
The smallest integers with $240$ divisors are:
 * $720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$

Proof
In the below, $\map {\sigma_0} n$ denotes the divisor counting function of $n$.

Then: