Corpus ID: 237504383

Menon-type identities concerning subsets of the set $\{1,2,\ldots,n\}$

  title={Menon-type identities concerning subsets of the set \$\\{1,2,\ldots,n\\}\$},
  author={L'aszl'o T'oth},
  • L. T'oth
  • Published 14 September 2021
  • Mathematics
d|n 1 is the divisor function. Identity (1.1) is due to P. K. Menon [6], and it has been generalized in various directions by several authors, also in quite recent papers. See, e.g., [1, 2, 3, 4, 5, 10, 11, 12, 13] and their references. For a nonempty subset A of {1, 2, . . . , n} let (A) denote the gcd of the elements of A. Then A is said to be relatively prime if (A) = 1, i.e., the elements of A are relatively prime. Let f(n) denote the number of relatively prime subsets of {1, 2, . . . , n… Expand
1 Citations
Proofs, generalizations and analogs of Menon's identity: a survey
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 aExpand


Affine invariants, relatively prime sets, and a phi function for subsets of .
A nonempty subset A of {1, 2, . . . , n} is relatively prime if gcd(A) = 1. Let f(n) and fk(n) denote, respectively, the number of relatively prime subsets and the number of relatively prime subsetsExpand
On the Number of Certain Relatively Prime Subsets of {1, 2, . . . , n}
  • L. Tóth
  • Computer Science, Mathematics
  • Integers
  • 2010
A fourth formula concerning the Euler-type function Φ k is generalized and certain related divisor-type, sum-of-divisors-type and gcd-sum-type functions are investigated. Expand
Menon’s identity with respect to a generalized divisibility relation
Summary.We present a generalization of P. Kesava Menon’s identity $$ {\sum\limits_{\scriptstyle a{\left( {\bmod {\rm{ }}n} \right)} \hfill \atop \scriptstyle (a,n) = 1 \hfill} {{\left( {a - 1,n}Expand
Menon-type identities with respect to sets of units
Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots ,u_m\in \mathbbExpand
Menon-type identities again: A note on a paper by Li, Kim and Qiao
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and alsoExpand
Menon-type identities concerning Dirichlet characters
Let χ be a Dirichlet character (mod n) with conductor d. In a quite recent paper Zhao and Cao deduced the identity ∑k=1n(k − 1,n)χ(k) = φ(n)τ(n/d), which reduces to Menon’s identity if χ is theExpand
The On-Line Encyclopedia of Integer Sequences
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  • Mathematics, Computer Science
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The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used. Expand
Menon's identity and arithmetical sums representing functions of several variables
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the directExpand
Another regular Menon-type identity in residually finite Dedekind domains
  • C. Ji, Y. Wang
  • Mathematics
  • 2020
We give a regular extension of the Menon-type identity to residually finite Dedekind domains.
A Menon-Sury-type identity for arithmetic functions on Fq[T
  • Publ. Math. Debrecen
  • 2021