Fungrim home page

Fungrim entry: 0fdb94

dnφ(d)nd=k=1ngcd ⁣(n,k)\sum_{d \mid n} \varphi(d) \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(d) \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(k) Sum over divisors
Totientφ(n)\varphi(n) Euler totient function
Sumnf(n)\sum_{n} f(n) 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)), For(d, n)), Sum(GCD(n, k), For(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

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

2019-12-11 23:01:54.699850 UTC