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} $.