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Fungrim entry: ea27a7

∑k=1nφ(k)⌊nk⌋=n(n+1)2\sum_{k=1}^{n} \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor = \frac{n \left(n + 1\right)}{2}
Assumptions:n∈Z≥0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{n} \varphi(k) \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
Sum∑nf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) Euler totient function
ZZGreaterEqualZ≥n\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))), For(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.

2021-03-15 19:12:00.328586 UTC