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Fungrim entry: 90bb4a

dnφ ⁣(d)d=(2nk=1nlcm ⁣(n,k))1\sum_{d \mid n} \varphi\!\left(d\right) d = \left(\frac{2}{n} \sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right)\right) - 1
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
\sum_{d \mid n} \varphi\!\left(d\right) d = \left(\frac{2}{n} \sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right)\right) - 1

n \in \mathbb{Z}_{\ge 0}
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
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
LCMlcm ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(DivisorSum(Mul(Totient(d), d), d, n), Sub(Parentheses(Mul(Div(2, n), Sum(LCM(n, k), Tuple(k, 1, n)))), 1))),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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2019-08-19 14:38:23.809000 UTC