$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$ Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$.
Are there any positive integer $n$ and negative integer $a$ such that $f_n(a)$ is an integer?