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 count function of $n$.

Then:

$\ds \map {\sigma_0} {720 \, 720}$

$=$

$\ds 240$

$\sigma_0$ of $720 \, 720$

$\ds \map {\sigma_0} {831 \, 600}$

$=$

$\ds 240$

$\sigma_0$ of $831 \, 600$

$\ds \map {\sigma_0} {942 \, 480}$

$=$

$\ds 240$

$\sigma_0$ of $942 \, 480$

$\ds \map {\sigma_0} {982 \, 800}$

$=$

$\ds 240$

$\sigma_0$ of $982 \, 800$

$\ds \map {\sigma_0} {997 \, 920}$

$=$

$\ds 240$

$\sigma_0$ of $997 \, 920$

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In particular: Prove that these are indeed the smallest
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Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $240$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $240$