Is there a designation for this not-quite-exponential time?
Two useful classes are sub-exponential, $c^{n^\epsilon}$ for $\epsilon \in (0,1)$, and quasi-polynomial, $c^{\log^k n}$ for $k &gt; 1$. (In both cases, $c &gt; 1$.) 

The terms themselves sometimes have different meanings. For example, in some contexts "sub-exponential" means $2^{(1-\epsilon)n}$ for some $\epsilon &gt; 0$, and in others it means $2^{o(n)}$.

Since the term "sub-exponential" is loaded, it is better to mention what you mean by it whenever you use it.