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Fungrim entry: 0fdb94

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

n \in \mathbb{Z}_{\ge 0}
Definitions:
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
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("0fdb94"),
    Formula(Equal(DivisorSum(Mul(Totient(d), Div(n, d)), d, n), Sum(GCD(n, k), Tuple(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

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2019-08-17 11:32:46.829430 UTC