Is there a closed-form for $$f(n)=\sum\limits_{\substack{k=1 \\ (k,n)=1}}^{n-1} \frac{k}{n-k}$$ For example, $f(5)=1/4+2/3+3/2+4/1= 6+5/12$; $f(6)=5+1/5$
The list of $f(x)$ from $x=1$ to $x=8$ is:
$(0,1,5/2,10/3,77/12,26/5,223/20,988/105)$
I'm trying to plot this but I would take a while to do it by hand that's why I ask. If there isn't a nice formula could you write a program that plots it I'm not the best at coding.