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Fungrim entry: aaef97

k=1ngcd ⁣(n,k)=dndφ ⁣(nd)\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \mid n} d \varphi\!\left(\frac{n}{d}\right)
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \mid n} d \varphi\!\left(\frac{n}{d}\right)

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
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("aaef97"),
    Formula(Equal(Sum(GCD(n, k), Tuple(k, 1, n)), DivisorSum(Mul(d, Totient(Div(n, d))), d, n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

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2019-08-17 11:32:46.829430 UTC