On a modification of $\underline{Set}(n)$

A modification of the set $\underline{\rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $\underline{\rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $\underline{\rm Set}(n,\varphi), \underline{\rm Set}(n,\psi), \underline{\rm Set}(n,\sigma)$ are discussed, where $\varphi, \psi$ and $\sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.