# Fungrim entry: a08583

$\varphi(n) \sigma_{0}\!\left(n\right) = \sum_{k=1}^{n} \begin{cases} \gcd\!\left(n, k - 1\right), & \gcd\!\left(n, k\right) = 1\\0, & \text{otherwise}\\ \end{cases}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
Menon's identity
TeX:
\varphi(n) \sigma_{0}\!\left(n\right) = \sum_{k=1}^{n} \begin{cases} \gcd\!\left(n, k - 1\right), & \gcd\!\left(n, k\right) = 1\\0, & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Totient$\varphi(n)$ Euler totient function
DivisorSigma$\sigma_{k}\!\left(n\right)$ Sum of divisors function
Sum$\sum_{n} f(n)$ Sum
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("a08583"),
Formula(Equal(Mul(Totient(n), DivisorSigma(0, n)), Sum(Cases(Tuple(GCD(n, Sub(k, 1)), Equal(GCD(n, k), 1)), Tuple(0, Otherwise)), For(k, 1, n)))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(0))),
Description("Menon's identity"))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC