Fungrim entry: aaef97

\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \mid n} d \varphi\!\left(\frac{n}{d}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \mid n} d \varphi\!\left(\frac{n}{d}\right)

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`DivisorSum`	$\sum_{k \mid n} f(k)$	Sum over divisors
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("aaef97"),
    Formula(Equal(Sum(GCD(n, k), For(k, 1, n)), DivisorSum(Mul(d, Totient(Div(n, d))), For(d, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

Greatest common divisor

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