The sequence \left(a_{n}\right) 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_{1} a_{2} \cdots a_{k} is an integer?
Answer Choices
A. 17
B. 18
C. 19
D. 20
E. 21