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What is the growth rate of the sum of powers of distinct primes closest to a given a integer?

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 $...
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