smc

First, substitute $2^{17}$ with $x$. Then, the given equation becomes $\frac{x^{17}+1}{x+1}=x^{16}-x^{15}+x^{14}...-x^1+x^0$ by sum of powers factorization. Now consider only $x^{16}-x^{15}$. This equals $x^{15}(x-1)=x^{15}\cdot (2^{17}-1)$. Note that $2^{17}-1$ equals $2^{16}+2^{15}+...+1$, by difference of powers factorization (or by considering the expansion of $2^{17}=2^{16}+2^{15}+...+2+2$). Thus, we can see that $x^{16}-x^{15}$ forms the sum of 17 different powers of 2. Applying the same method to each of $x^{14}-x^{13}$, $x^{12}-x^{11}$, ... , $x^2-x^1$, we can see that each of the pairs forms the sum of 17 different powers of 2. This gives us $17\cdot 8=136$. But we must count also the $x^0$ term. Thus, Our answer is $136+1=\boxed{137}$.