$\Large a_0 = 1, a_1 = \sqrt[19]{2}, \\ \Large a_n = a_{n-1} \times a_{n-2} ^2$

Consider a sequence defined recursively for $n\geq 2$ in the above manner.

What is the smallest positive integer value $k$ such that the product $a_{1}a_{2}\dotsm a_k$ is an integer?