This can be factored as $(2^8)(1 + 8 + 2^{n - 8})$. Since $2^8$ is a perfect square, $9 + 2^{n - 8}$ must be a perfect square. Let this perfect square be $a^2$. Then, $a^2 - 9 = 2^{n - 8}$ which is a power of $2$, so $(a + 3)(a - 3)$ must be a power of $2$. Hence, $a$ is odd, and the difference between 2 powers of $2$ is $6$. We see that only $2$ and $8$ follow this, so $a + 3 = 8$ and $a - 3 = 2$, meaning that $a = 5$ and $n = 12$. This is only for $n \geq 8$; we see that there are no values $n < 8$ for which this is satisfied so $\boxed{12}$