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

Totient function