A Generalization of Euler's Totient Function

J. Freed-Brown, M. Holder, M. E. Orrison and M. Vrable introduced a new generalization of Euler's totient function, $M_k(n)$, defined to be the number of sequences $(g_1,\dots,g_k)$ of elements in $\mathbb{Z}_n$ such that if $G_i$ is the subgroup of $\mathbb{Z}_n$ generated by $\{g_1,\dots,g_i\}$, then \[\{0\}<G_1<\dots<G_{k-1}<G_k=\mathbb{Z}_n.\] They also defined the function $M(n)$ to be $M_k(n)$ where $k$ is the largest integer such that $M_k(n)$ is nonzero. They gave the formulas for $M_k(p^e)$ and $M(p^eq^f)$ where $p$ and $q$ are distinct primes and $k,e$ and $f$ are natural numbers. In this article, some more properties of $M_k(n)$ and $M(n)$ are investigated.