Another generalization of Euler's arithmetic function and Menon's identity

We define the $k$-dimensional generalized Euler function ${\phi }_k(n)$ as the number of ordered $k$-tuples $(a_1,\dots ,a_k)\in {\mathbb{N}}^k$ such that $1\le a_1,\dots ,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$. We investigate some of properties of the function ${\phi }_k(n)$, and obtain a corresponding Menon-type identity.