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Fungrim entry: a05466

dnφ ⁣(d)σk ⁣(nd)=nσk1 ⁣(n)\sum_{d \mid n} \varphi\!\left(d\right) \sigma_{k}\!\left(\frac{n}{d}\right) = n \sigma_{k - 1}\!\left(n\right)
Assumptions:kZ1andnZ0k \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{d \mid n} \varphi\!\left(d\right) \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
DivisorSumknf ⁣(k)\sum_{k \mid n} f\!\left(k\right) Sum over divisors
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
ZZGreaterEqualZn\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))), d, n), Mul(n, DivisorSigma(Sub(k, 1), n)))),
    Variables(k, n),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC