I'm interested in asymptotics for the sum
$$\sum\_{n\le x}\frac{n}{\text{rad}(n)}.$$
For my research I only need to know whether or not this is $\mathcal{O}(x)$, but I would appreciate more precise asymptotics. Additionally, a reference would be much appreciated. The sum of $\frac{1}{\text{rad}(n)}$ has been discussed on the site, but I cannot find anything on my sum.



Asymptotics on sum of n/rad(n)
One has
\begin{align\*}
\sum\_{n \leq x}\frac{n}{\operatorname{rad}n} & = (1+o(1)) \, x \sqrt{\frac{2}{\log x \log \log x}} \exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty), \\
& = x\exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty) \\
& \neq O(x\, (\log x)^A) \ (x\to \infty)
\end{align\*}


for any $A \in \mathbb{R}$, where the first estimate holds according to user "Ofir Gorodetsky"'s detailed answer to a similar question to yours at [Asymptotic behavior of a "strange" arithmetic function](https://mathoverflow.net/questions/445395/asymptotic-behavior-of-a-strange-arithmetic-function)


