On certain sums involving the largest prime factor over integer sequences

Given an integer $n\ge 2$, its prime factorization is expressed as $n={\prod }_{i=1}^s{p}_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to derive an asymptotic formula for both sums ${\sum }_{n\le x}f(n)$ and ${\sum }_{n\le x,n\in S_k}f(n)$, where $S_k$ denotes the set of all $k$-free integers.