AMC 12 2020a Problem 19

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that$$\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.$$What is $k?$

$\textbf{(A) } 117 \textbf{(B) } 136 \textbf{(C) } 137 \textbf{(D) } 273 \textbf{(E) } 306$