I have come across a multiplicative function (almost always named "$\psi$") defined on prime powers by $$\psi(p^n) = (p^n - 1)(p^{n-1} - 1) \cdots (p-1),$$ for primes $p$ and integers $n\geq 0$. (So, $\psi(1)=1$ and $\psi(ab) = \psi(a)\psi(b)$, for relatively prime positive integers $a$ and $b$). However, no author I've read so far has stated a non-symbolic name for this, and I didn't get anything close to this using Google.
Does this $\psi$ have a name? Is it something well-studied in number theory?
(Note that it coincides with Euler's $\phi$ on square-free positive integers.)
Context. This function arises in characterising orders of finite groups with certain properties.