Let our positive integers be $ a $ and $ b $. It follows that \[ab - (a + b) = ab - a - b = 39.\]The left-hand side reminds us of Simon's Favorite Factoring Trick. So, we add $ 1 $ to both sides to "complete the rectangle": \[ab - a - b + 1 = 39 + 1 \implies (a - 1)(b - 1) = 40.\]We can narrow down the possibilities by considering all the positive factor pairs of $ 40 $. We disregard values of $ a $ and $ b $ that are not relatively prime positive integers both less than $ 20 $. \begin{array}{c|c|c|c|c} a-1 & b-1 & a & b & \text{Feasible} \\ \hline 1 & 40 & 2 & 41 & \times \\ \hline 2 & 20 & 3 & 21 & \times \\ \hline 4 & 10 & 5 & 11 & \checkmark \\ \hline 5 & 8 & 6 & 9 & \times \end{array}The only possibility that works is $ a = 5 $ and $ b = 11 $, or, by symmetry, $ a = 11 $ and $ b = 5 $. Either way, the sum $ a + b $ is equal to $ \boxed{16} $.