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Fungrim entry: c24323

k=1nlcm ⁣(n,k)=n2(1+dndφ ⁣(d))\sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right) = \frac{n}{2} \left(1 + \sum_{d \mid n} d \varphi\!\left(d\right)\right)
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right) = \frac{n}{2} \left(1 + \sum_{d \mid n} d \varphi\!\left(d\right)\right)

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
LCMlcm ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
DivisorSumknf ⁣(k)\sum_{k \mid n} f\!\left(k\right) Sum over divisors
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("c24323"),
    Formula(Equal(Sum(LCM(n, k), Tuple(k, 1, n)), Mul(Div(n, 2), Add(1, DivisorSum(Mul(d, Totient(d)), d, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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2019-08-19 14:38:23.809000 UTC