For every
![n\in\mathbb{N}](/media/m/2/8/1/281f646b448641f8943c8ee1e9772add.png)
let
![d(n)](/media/m/8/b/5/8b5ba2b86903af1640ec9f08b90773b6.png)
denote the number of (positive) divisors of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
. Find all functions
![f: \mathbb{N}\to\mathbb{N}](/media/m/d/a/7/da73b04c80e08543343ab5b33ffc9c67.png)
with the following properties:
![d\left(f(x)\right) = x](/media/m/a/9/0/a908f145b1eca2e9cf677bb5b30ad752.png)
for all
![x\in\mathbb{N}](/media/m/2/f/9/2f98be09ec6c0bb8b292a2ff8f8d78bc.png)
.
![f(xy)](/media/m/0/8/b/08b7ea286ad996f88e36e0473893a424.png)
divides
![(x - 1)y^{xy - 1}f(x)](/media/m/2/3/f/23f1adbacd528349c72f2d86c83e9317.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y\in\mathbb{N}](/media/m/e/5/c/e5c8eb0b91e5bbffc5fb7657fac3182d.png)
.
Proposed by Bruno Le Floch, France
%V0
For every $n\in\mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ with the following properties: $d\left(f(x)\right) = x$ for all $x\in\mathbb{N}$. $f(xy)$ divides $(x - 1)y^{xy - 1}f(x)$ for all $x$, $y\in\mathbb{N}$.
Proposed by Bruno Le Floch, France