AMC 10A/12A 2020 (Problem 21/19)

There exists a unique strictly increasing sequence of nonnegative integers

a_1<a_2<\cdots<a_k

such that

\frac{2^{289}+1}{2^{17}+1}=2^{a_1}+2^{a_2}+\cdots+2^{a_k}

. What is

k

\text{(A)}\;117

\text{(B)}\;136

\text{(C)}\;137

\text{(D)}\;273

\text{(E)}\;306