Does this function have a "name", somewhat linked to Euler totient

Question

If $\varphi$ denotes the Euler totient and $n=p_1^{k_1}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ we have

$ \varphi(n)= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})= p_1^{k_1} \left(1- \frac{1}{p_1} \right) p_2^{k_2} \left(1- \frac{1}{p_2} \right) \cdots p_r^{k_r} \left(1- \frac{1}{p_r} \right)$.

I using a function defined by $\psi(n) = \varphi(p_1^{k_1}) + \varphi(p_2^{k_2})+ \cdots + \varphi(p_r^{k_r})$.

I am wondering if this function has a "classical name" that I should use (and maybe another well-known notation than $\psi$).

Thank you!

Answer 1

I doubt this function has a classical name or notation. It is an example of an "additive function", which means that $\psi(ab) = \psi(a) + \psi(b)$ whenever $\gcd(a,b)=1$.