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Fungrim entry: a05466

dnφ(d)σk ⁣(nd)=nσk1 ⁣(n)\sum_{d \mid n} \varphi(d) \sigma_{k}\!\left(\frac{n}{d}\right) = n \sigma_{k - 1}\!\left(n\right)
Assumptions:kZ1  and  nZ0k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
\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}
Fungrim symbol Notation Short description
DivisorSumknf(k)\sum_{k \mid n} f(k) Sum over divisors
Totientφ(n)\varphi(n) 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:
    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)))))

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2021-03-15 19:12:00.328586 UTC