Fungrim entry: 90bb4a

\sum_{d \mid n} \varphi(d) d = \left(\frac{2}{n} \sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right)\right) - 1

Assumptions:

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

TeX:

\sum_{d \mid n} \varphi(d) d = \left(\frac{2}{n} \sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right)\right) - 1

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

Definitions:

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

Source code for this entry:

Entry(ID("90bb4a"),
    Formula(Equal(DivisorSum(Mul(Totient(d), d), For(d, n)), Sub(Parentheses(Mul(Div(2, n), Sum(LCM(n, k), For(k, 1, n)))), 1))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

Totient function

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