Consider the following for $n\geq1$:

$$S_n = \sum_{i=1}^n i^{\ln(i)}$$

It seems to be hard to give an exact closed form formula for $S_n$.

It is straightforward however to see that $n^{\ln(n)} \leq S_n \leq n^{\ln(n)+1}$ but it is not clear to me if $S_n$ is asymptotic to some constant times the lower or upper bound or some function in between.

What is the asymptotic growth of $S_n$?

I would be interested in a $\Theta()$ answer if finding the precise constants is hard.

My motivation is mostly mathematical curiosity. I don't have a real application that isn't covered by the simple bounds given above.

(Much the same question asked previously on math.se.)