2020 AMC 10A Problem 21

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

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

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