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Fungrim entry: 90bb4a

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

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
DivisorSumknf(k)\sum_{k \mid n} f(k) Sum over divisors
Totientφ(n)\varphi(n) Euler totient function
Sumnf(n)\sum_{n} f(n) 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:
Entry(ID("90bb4a"),
    Formula(Equal(DivisorSum(Mul(Totient(d), d), For(d, n)), Sub(Parentheses(Mul(Div(2, n), Sum(LCM(n, k), For(k, 1, n)))), 1))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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2019-11-19 15:10:20.037976 UTC