A Menon-type identity using Klee's function

Chandran, Arya; Thomas, Neha Elizabeth; Namboothiri, K Vishnu

doi:10.21136/cmj.2021.0370-20

Cited by 3 publications

(8 citation statements)

References 12 publications

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“…Here the inner sum is φ(n)/φ(d) by Lemma 1, which proves identity (7). In the case f = id we obtain (1).…”

Section: Methods Iv: a Variant Of Methods IIIsupporting

confidence: 56%

“…where g = id φ * 1, i.e., g(n) = d|n dφ(d). Here (39) immediately follows from identity (7). Menon also proved the following result, related to the gcd-sum function (4).…”

Section: Remarks On the Paper By Menonmentioning

confidence: 81%

“…therefore ( 32), ( 33) and (34) imply formula (7), which reduces to Menon's original identity in the case f = id.…”

Section: Tóthmentioning

confidence: 92%

“…For an arbitrary function f, identity (7) was proved by Sita Ramaiah [67, Th. 9.1] in a more general form, concerning regular systems of divisors (see Section 4.9) and k-reduced residue systems (see Section 4.6).…”

Section: Some More Remarksmentioning

confidence: 99%

“…For example: How to modify identity (1) if (a − 1, n) is replaced by f((a − 1, n)), where f is an arithmetic function, or by (P(a), n), where P is an arbitrary polynomial with integer coefficients, in particular P(a) = a − s, with s a given integer? As answers to these questions see identities (7), (32), (52), (48) and (53).…”

Section: Introduction 1motivationmentioning

confidence: 99%

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Proofs, generalizations and analogs of Menon’s identity: a survey

Tóth

2023

Acta Universitatis Sapientiae, Mathematica

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Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.

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“…Here the inner sum is φ(n)/φ(d) by Lemma 1, which proves identity (7). In the case f = id we obtain (1).…”

Section: Methods Iv: a Variant Of Methods IIIsupporting

confidence: 56%

“…where g = id φ * 1, i.e., g(n) = d|n dφ(d). Here (39) immediately follows from identity (7). Menon also proved the following result, related to the gcd-sum function (4).…”

Section: Remarks On the Paper By Menonmentioning

confidence: 81%

“…therefore ( 32), ( 33) and (34) imply formula (7), which reduces to Menon's original identity in the case f = id.…”

Section: Tóthmentioning

confidence: 92%

Section: Some More Remarksmentioning

confidence: 99%