Function with limiting behavior between polynomial and exponential?

Question

Does there exist a function $f$ such that

For all $p>0$, $$\lim_{n \to \infty}\frac {|f(n)|}{|n^p|}>1$$ And

For all $a>0$, $$\lim_{n \to \infty}\frac {|f(n)|}{|e^{an}|}<1$$

Answer 1

$$ e^{\sqrt n} $$

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