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Fungrim entry: ea27a7

k=1nφ ⁣(k)nk=n(n+1)2\sum_{k=1}^{n} \varphi\!\left(k\right) \left\lfloor \frac{n}{k} \right\rfloor = \frac{n \left(n + 1\right)}{2}
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{n} \varphi\!\left(k\right) \left\lfloor \frac{n}{k} \right\rfloor = \frac{n \left(n + 1\right)}{2}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
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("ea27a7"),
    Formula(Equal(Sum(Mul(Totient(k), Floor(Div(n, k))), Tuple(k, 1, n)), Div(Mul(n, Add(n, 1)), 2))),
    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.

2019-08-17 11:32:46.829430 UTC