1987 AMC 12 Question 29

Consider the sequence of numbers defined recursively by $t_1=1$ and for $n\gt1$ by $t_n=1+t_{\frac{n}{2}}$ when $n$ is even and by $t_n=\frac{1}{t_{n-1}}$ when $n$ is odd. Given that $t_n=\frac{19}{87},$ the sum of the digits of $n$ is