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
$$f(n):=\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\rfloor\right)},\tag{1}$$ where $\lfloor x\rfloor$ is the floor function and $\operatorname{rad}(n)$ is the squarefree part of $n$, see this Wikipedia, and
$$g(n):=\sum_{k\mid n}k^{\operatorname{rad}\left(\frac{n}{k}\right)}.\tag{2}$$
Question. I would like to know a way to do a comparison of the sizes of the arithmetic functions defined previously when $n$ is large. What's a a good way to do it? Many thanks.
The corresponding sequence to the arithmetic function defined in $(1)$ starts as $1, 3, 6, 14, 17, 33\ldots$ and for $(2)$ as $1, 3, 4, 9, 6, 24, 8, 29\ldots$ Make the comparison with A055225 from the OEIS. I believe thus that these sequences or arithmetic functions were not in the literature.
If your approach is using asymptotics or means, feel free to explain me your strategy to get an idea how much different are the sizes of our functions.