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

Totient function