# Fungrim entry: a05466

$\sum_{d \mid n} \varphi(d) \sigma_{k}\!\left(\frac{n}{d}\right) = n \sigma_{k - 1}\!\left(n\right)$
Assumptions:$k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\sum_{d \mid n} \varphi(d) \sigma_{k}\!\left(\frac{n}{d}\right) = n \sigma_{k - 1}\!\left(n\right)

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; 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
DivisorSigma$\sigma_{k}\!\left(n\right)$ Sum of divisors function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("a05466"),
Formula(Equal(DivisorSum(Mul(Totient(d), DivisorSigma(k, Div(n, d))), For(d, n)), Mul(n, DivisorSigma(Sub(k, 1), n)))),
Variables(k, n),
Assumptions(And(Element(k, ZZGreaterEqual(1)), 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