Define a function $f: \mathbb N \to \mathbb N$ by $f(p^k) = p^k-1$ on prime powers and extended to full natural numbers by defining $f$ to be a multiplicative function. Then $f$ agrees with the Euler-totient function on the set of square-free numbers. So I believe the function $f$ is somewhat natural.
My question is, whether $f$ is a well-known arithmetic function? If not, can it be written as a product of well-known arithmetic functions? Kindly share your thoughts. Thank you.