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Fungrim entry: aaef97

k=1ngcd ⁣(n,k)=d{1,2,n},dndφ ⁣(nd)\sum_{k=1}^{n} \gcd\!\left(n, k\right) = \sum_{d \in \{1, 2, \ldots n\},\,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 \in \{1, 2, \ldots n\},\,d \mid n} d \varphi\!\left(\frac{n}{d}\right)

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
GCDgcd ⁣(n,k)\gcd\!\left(n, k\right) Greatest common divisor
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
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)), SumCondition(Mul(d, Totient(Div(n, d))), d, And(Element(d, ZZBetween(1, n)), Divides(d, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

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2019-06-18 07:49:59.356594 UTC