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Lower bound related to the number of distinct prime numbers
Let $\omega(n)$ be the number of distinct prime factors of $n$ (without multiplicity, of course). I know some results about average of $\omega(n)$. But I didn't find any result about the following: ...
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Compare $\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\rfloor\right)}$ and $\sum_{k\mid n}k^{\operatorname{rad}\left(\frac{n}{k}\right)}$
I would like to know how do a comparison between the sizes of these functions defined for integers $n\geq 1$, when $n$ is large
$$f(n):=\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\...
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