Smallest Triplet of Integers whose Product with Divisor Count are Equal

Theorem
Let $\map {\sigma_0} n$ denote the divisor count function: the number of divisors of $n$.

The smallest set of $3$ integers $T$ such that $m \, \map {\sigma_0} m$ is equal for each $m \in T$ is:
 * $\set {168, 192, 224}$