Fungrim entry: aaef97

\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \in \{1, 2, \ldots n\},\,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 \in \{1, 2, \ldots n\},\,d \mid n} d \varphi\!\left(\frac{n}{d}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`ZZBetween`	$\{a, a + 1, \ldots b\}$	Integers between a and b inclusive
`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), Tuple(k, 1, n)), SumCondition(Mul(d, Totient(Div(n, d))), d, And(Element(d, ZZBetween(1, n)), Divides(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.

2019-06-18 07:49:59.356594 UTC