Let $n$ be a positive integer, and $$2 = p_1 < p_2 < \dots < p_m \le n$$ be the sequence of all primes less than or equal to $n$.

For each index $j$ let $p_j^{e_j}$ be the largest power of $p_j$ still less than or equal to $n$. Define

$$S_n = p_1^{e_1} + p_2^{e_2} + \dots + p_{m}^{e_m} $$

to be the sum of these prime powers. What is the growth rate of this series $S_n$?

We can get an upper bound of

$$S_n \le nm \sim \frac{n^2}{\ln n}$$

by the prime number theorem, and a lower bound of

$$S_n \ge \lfloor n/p_1\rfloor + \lfloor n/p_2\rfloor + \dots + \lfloor n/p_m\rfloor \sim n\ln \ln n $$

using the fact that each term $p_j^{e_j}$ is within a factor of $p_j$ of $n$ and asymptotics for the sum of the reciprocals of the primes.

However, there's still some gap between these two bounds. Is the precise asymptotic growth rate of $S_n$ known?