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Fungrim entry: 93a877

φ(n)=k=1ngcd ⁣(n,k)cos ⁣(2πkn)\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) \cos\!\left(\frac{2 \pi k}{n}\right)
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) \cos\!\left(\frac{2 \pi k}{n}\right)

n \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
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
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Totient(n), Sum(Mul(GCD(n, k), Cos(Div(Mul(Mul(2, Pi), k), n))), For(k, 1, n)))),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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2021-03-15 19:12:00.328586 UTC