We denote the main/principal branch of the Lambert $W$ function as $W(x)$, and we define the multiplicative function $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\ \prod_{p\mid n}e^{-W(p^{e_p})}, & \text{if $n>1$ and has canonical representation $n=\prod_{p\mid n}p^{e_p}$} \end{cases}.$$
Wikipedia has the article Lambert W function dedicated to this function, and a section dedicated to the canonical representation of a positive integer in the article Fundamental theorem of arithmetic.
I don't know if this function is in the literature, if this is the case please answer as a reference request my question, and I can to search and read these facts from the literature.
Question. I would like to know if we can to state some relevant asymptotic for $$\sum_{1\leq n\leq x}f(n)$$ as $x$ grows to $\infty$ (this is the approach to study certain arithmetic functions, see for example the introduction of chapter 3 Averages of Arithmetical Function from [1]). Many thanks.
That I wanted to ask with previous question is what about the size of our function $f(n)$, since it fluctuates it is better to take averages of particular values of $f(n)$. If my calculations were right, but these aren't adressing the question, I can to deduce that 1) our function can be represented for integers $n>1$ as $$f(n)=\frac{1}{n}\left(\prod_{p\mid n}\log(p^{e_p})\right)\cdot e^{o(1)\omega(n)}\tag{1}$$ where $\omega(n)$ denotes the prime omega function $\sum_{p\mid n}1$, and 2) that one has in Vinogradov notation that $$f(n)\ll n^{\delta-1}e^{o(1)\omega(n)}\tag{2}$$ for each $\delta>0$. Both statements deduced/inspired from formulas of the linked Wikipedia, if I'm right. From the representation $(1)$ we can deduce also an inequality $f(n)<\text{Bound}(n)$ invoking the inequality of arithmetic and geometric means, with $\text{Bound}(n)$ expressed, in particular, in terms of particular values of the prime omega function.
But I need a summation method and calculations to get an answer for the question, I don't know if previous calculations can be helpful.
Remarks. I doubt that previous multiplicative function has the applications that I had in mind (when I tried to evoke combinations of certain diophantine equations and an identity that involves $W(x)$ for reals $x>0$), but I think that to know a relevant/remarkable asymptotic identity for $\sum_{1\leq n\leq x}f(n)$ can be interesting.
References:
[1] Tom M. Apostol, Introduction to Analytic Number Theory, Springer (1976).
[2] A. Hoorfar and M. Hassani, Inequalities on the Lambert W Function and Hyperpower Function, JIPAM, Volume 9, Issue 2, Article 51 (2008).