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

We define the k-dimensional generalized Euler function $$\varphi _k(n)$$ as the number of ordered k-tuples $$(a_1,\ldots ,a_k)\in {\mathbb {N}}^k$$ such that $$1\le a_1,\ldots ,a_k\le n$$ 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 the properties of the function $$\varphi _k(n)$$ , and obtain a corresponding Menon-type identity.