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Fungrim entry: c24323

∑k=1nlcm⁡ ⁣(n,k)=n2(1+∑d∈{1,2,…n}, d∣ndφ ⁣(d))\sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right) = \frac{n}{2} \left(1 + \sum_{d \in \{1, 2, \ldots n\},\,d \mid n} d \varphi\!\left(d\right)\right)
Assumptions:n∈Z≥0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{n} \operatorname{lcm}\!\left(n, k\right) = \frac{n}{2} \left(1 + \sum_{d \in \{1, 2, \ldots n\},\,d \mid n} d \varphi\!\left(d\right)\right)

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
LCMlcm⁡ ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
ZZBetween{a,a+1,…b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
ZZGreaterEqualZ≥n\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, SumCondition(Mul(d, Totient(d)), d, And(Element(d, ZZBetween(1, n)), Divides(d, n))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC