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

Abstract

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