# 2016 AMC 12B Problem 25

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

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?

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