Example of sequence with interesting rate of growth

Question

I am looking for an example of a sequence $\{a_n\}_{n\in \mathbb{N}}$, with $a_n \geq 0$, such that for $k<2$ $$ \limsup_{n\to \infty} \frac{a_n}{n^k} >0 ,$$ and that $$ \limsup_{n\to \infty} \frac{a_n}{n^2} = 0 .$$

I do not know if this is possible, so if you happen to know at least why this behavior might be possible or not that would help too.

Answer 1

Take $a_n=\frac{n^2}{\log n}$. Then $\limsup_n\frac{a_n}{n^k}=\infty$ if $k<2$ and $\limsup_n\frac{a_n}{n^2}=0$.