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

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