Does there exist an integer sequence $\{a_n\}_{n = 1}^\infty$ that satisfies the following properties:
- $\forall t > 1, n^t = o(a_n)$
- $\forall p > 1, q > 0, a_n = o(p^{n^q})$ ?
The only thing I managed to determine, was, that the convergence radius of its generating function is $1$ and thus, by Pringsheim theorem it has a singularity in $1$, which is essential, because if it were a pole of degree $k$, then it would have meant, that $a_n = O(n^k)$, which is certainly not true.
However, that does not seem to be much helpful.