Let $f:\mathit{N}\to \mathit{R}$ be a function such that $f\left(x+y\right)=2 f\left(x\right) f\left(y\right)$ for natural numbers $x$ and $y$. If $f\left(1\right)=2$, then the value of $\alpha$ for which $\sum _{k=1}^{10}f\left(\alpha +k\right)=\frac{512}{3}\left(2^{20}-1\right)$ holds, is