AMC 12 2016b Problem 25

The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?

\textbf{(A)}\ 17 \textbf{(B)}\ 18 \textbf{(C)}\ 19 \textbf{(D)}\ 20 \textbf{(E)}\ 21