A Menon-type identity using Klee's function

TL;DR

This paper generalizes Menon's identity by incorporating Klee's function and a generalized gcd, extending previous results and providing a broader mathematical framework for gcd sum identities.

Contribution

It introduces a Menon-type identity involving Klee's function and a generalized gcd, expanding the scope of classical gcd sum identities.

Findings

Derived a new Menon-type identity using Klee's function

Generalized previous identity by Lee and Kim

Extended classical gcd sum identities to broader functions

Abstract

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. A natural generalization of $\phi$ is the Klee's function $\Phi_s$. In this paper we derive a Menon-type identity using Klee's function and a generalization of the gcd function. This identity generalizes an identity given by Lee and Kim in [\textit{J. Number Theory 175, 42--50(2017)}].