2016 AMC 12B 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\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 21

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.