HMMT 2019 A4

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying

• $f(1)=1$;
• for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$.

Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.