Limit of an unusual function?

Question

First, I define the function, $f(k)=1$ if the sum of the factors of $k$ (excluding $k$) is greater than $k$, $f(k)=-1$ if the sum of the factors of $k$ (excluding $k$) is less than $k$, and $f(k)=0$ if $k$ is a perfect number for $k\in\mathbb{N}$. Is it possible to know the value of the limit $$\lim_{n\rightarrow\infty}\frac{\sum_{k=1}^nf(k)}{n}?$$If so, what is it and how does one calculate it?

Answer 1

It is an old result of Davenport that the set of abundant numbers has a natural density. That natural density is now known to be somewhere between $0.2474$ and $0.2480$. Please see this paper by Marc Deléglise.

Since the perfect numbers have natural density $0$, your limit exists, and one can give a good estimate for its value.