Fungrim home page

Fungrim entry: a08583

φ(n)σ0 ⁣(n)=k=1n{gcd ⁣(n,k1),gcd ⁣(n,k)=10,otherwise\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:nZ0n \in \mathbb{Z}_{\ge 0}
Menon's identity
\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}
Fungrim symbol Notation Short description
Totientφ(n)\varphi(n) Euler totient function
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
Sumnf(n)\sum_{n} f(n) Sum
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    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)))),
    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.

2021-03-15 19:12:00.328586 UTC