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# 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

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