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?



What is the growth rate of the sum of powers of distinct primes closest to a given a integer?
A better lower bound is $S(n)$, the sum of all primes below $n$, and this lower bound makes a good asymptotic value. One can tweak this by observing that for every term corresponding to a prime less than $\sqrt{n}$ that term is at least $n^{2/3}$, so a tighter lower bound like $S(n) - S(\sqrt{n}) + n^{7/6}/\log n$ is available. Since $S(n)$ is like $O(n^2/\log n)$, one wonders how good an asymptotic is desired.


Gerhard "Wonders What This Is For" Paseman, 2020.08.10.


