, 2022

Solution:

For a positive integer $n$, let $\operatorname{rad} n$ be the product of the distinct prime factors of $n$. Observe that if $n \mid m^{m}$, all prime factors of $n$ must divide $m$, so $\operatorname{rad} n \mid m$.

Therefore, if $n$ is such an integer, $\operatorname{rad} n$ must divide at least two of the numbers in $\{1,2,3,4,5,6,7,8,9,10\}$, implying that $\operatorname{rad} n$ is either $1,2,3$, or $5$. These have $1,10,6$, and $5$ cases, respectively, for a total of $22$.