Divisors mu Möbius

Möbius inversion formula states that if $g$ and $f$ are arithmetic functions satisfying

$g\left(n\right)=\sum _{d|n}f\left(d\right)$

for every integer $n\ge 1$, then

$f\left(n\right)=\sum _{d|n}\mu \left(d\right)g\left(\frac{n}{d}\right)$

for every integer $n\ge 1$, where $\mu \left(d\right)$ is the Möbius function. Phi Euler's function satisfies

$n=\sum _{d|n}\phi \left(d\right)$

Then, applying Möbius inversion formula we can write

$\phi \left(n\right)=\sum _{d|n}\mu \left(d\right)\frac{n}{d}=n\sum _{\begin{array}{c}d|n\\ \mu \left(d\right)\ne 0\end{array}}\frac{1}{\mu \left(d\right)·d}$

divisors_mu_moebius(Integer)

Description

Given an integer $n$, returns a list with the divisors of $n$ multiplied by $\mu(d)$.