Let $\phi_k(n)$ be the number of integers $m$ in $1\le m\le n$ for which $\gcd(m,n) = k$. Then $\phi_1(n) =\varphi(n)$, the standard totient function.
This function arises in the analysis of the structure of the cyclic group of order $n$: the number of elements of $Z_n$ that have order $k$ is $\phi_k(n)$. It similarly arises in the analysis of the dihedral groups, which is where I ran into it. So I imagine that it has a standard name.
It is easy enough to calculate, but I would like to research it and I don't know what it is called.