jmc

The positive integers with exactly four positive factors can be written in the form $pq$, where $p$ and $q$ are distinct prime numbers, or $p^3$, where $p$ is a prime number.

Using this, we can see that the smallest five positive integers with exactly four positive factors are $2\cdot 3=6$, $2^3=8$, $2\cdot 5=10$, $2\cdot 7=14$, and $3\cdot 5=15$. Summing these numbers, we get $6+8+10+14+15=\boxed{53}$.