# Does there exist an integer sequence that satisfies the following properties?

Does there exist an integer sequence $$\{a_n\}_{n = 1}^\infty$$ that satisfies the following properties:

1. $$\forall t > 1, n^t = o(a_n)$$
2. $$\forall p > 1, q > 0, a_n = o(p^{n^q})$$ ?

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.

• $\lfloor n^{\log n}\rfloor$ should work. – Wojowu Apr 17 at 14:31
• @Wojowu, yes, this one indeed seems to work. You might probably want to post this as an answer, so I could accept it... – Yanior Weg Apr 17 at 15:46