Haukkanen, Pentti; Tóth, László Menon-type identities again: a note on a paper by Li, Kim and Qiao. (English) Zbl 1463.11003 Publ. Math. Debr. 96, No. 3-4, 487-502 (2020). Here is the authors’ abstract: “We give common generalizations of the Menon-type identities by R. Sivaramakrishnan [J. Indian Math. Soc., New Ser. 33, 127–132 (1969; Zbl 0206.33406)] and Y. Li et al. [Publ. Math. 94, No. 3–4, 467–475 (2019; Zbl 1438.11003)]. Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character-free approach for the proof.”For example, the authors present a formula for \[\sum_{1\le a_1\le n}\cdots \sum_{1\le a_k\le n}[(a_1-1,n),\dots, (a_k-1,n)],\] which for \(k=1\) yields Menon’s classical identity: \[M(n):=\sum_{a=1}^n (a-1, n) =\varphi(n) \tau(n). \] Here \(n\) and \(k\) are positive integers, \((\ ,\,)\) denotes the greatest common divisor, \([\ , \ ]\) denotes the least common multiple, \(\varphi\) is Euler’s totient function, and \(\tau(n)\) is the sum of the divisors of \(n\). Reviewer: Moshe Roitman (Haifa) Cited in 3 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11A25 Arithmetic functions; related numbers; inversion formulas Keywords:Menon’s identity; gcd-sum function; arithmetic function of several variables; multiplicative function; lcm function; polynomial with integer coefficients Citations:Zbl 0206.33406; Zbl 1438.11003 PDFBibTeX XMLCite \textit{P. Haukkanen} and \textit{L. Tóth}, Publ. Math. Debr. 96, No. 3--4, 487--502 (2020; Zbl 1463.11003) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Pillai’s arithmetical function: Sum_{k=1..n} gcd(k, n). a(n) = d(n) * phi(n), where d(n) is the number of divisors function.