If $f,g$ are multiplicative functions (with bell series $F_p(x), G_p(x)$) so is $n \mapsto f(n)/g(n)$, what is its Bell series? (or is there no nice way to write it in terms of them?)
I think it's not possible since it's composition in the case of completely multiplicative functions. I just want to know bell series for $\frac{n}{\varphi(n)}$ and $\sum_{d|n} \frac{\mu(d)^2}{\varphi(d)}$.
For the first I got $1 + \frac{p}{p-1}\cdot\frac{1}{1-x}$ and for the second $\tfrac{1}{1-x}\left(1 + \frac{1}{p-1}\right)$ so again I have this wrong.