Inspired Compound Figure

This is not a complete solution.

It was shown in the solutions to Immensely Compound Figure that $7!$ has $60$ divisors, but there are no smaller positive integers also with $60$ divisors. A similar method can be used to show that this property is true for $1 \leq n \leq 7$ .

However, for $n = 8$ , $8! = 2^7 \cdot 3^2 \cdot 5 \cdot 7$ and therefore has $(7 + 1)(2 + 1)(1 + 1)(1 + 1) = 96$ divisors. There are $5$ positive integers smaller than $8! = 40320$ that also have $96$ divisors: $36960$ , $27720$ , $37800$ , $30240$ , and $32760$ , and the sum of these numbers is $\boxed{165480}$ .