For every
let
denote the number of (positive) divisors of
. Find all functions
with the following properties:
for all
.
divides
for all
,
.
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
Školjka