jmc

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":

$$ab-a-b+1=39+1⟹(a-1)(b-1)=40.$$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} $a-1$ & $b-1$ & $a$ & $b$ & 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 $\boxed{16}$.