**Edit 01**:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, by wolfram alpha i have got no integer n>1 satisfies the equation $\sigma_y(n)\bmod \sigma_x(n)=0$ at a least it is true from $ n=2$ to $ 5000$ with $y=3$ and $x=2$,According to "Gerhard paseman Idea " we have the following question show the counter example and the problem I challenge is to find integers:$x,p, q ,y$ which presnt the counter example of the precedent equation,

Then i have tried to find these integers $x,p, q ,y$ which satisfy the following conditions but I don't succeed:

$p, q$ are primes, $x$ is a positive integer such that:

$(01)\quad $ $b$ is not a multiple of $x$, but $(1+p^x) $ divides $(1+q^b)$.

$(02)\quad$ $c$ is not a multiple of $x$, but $(1+q^x)$ divides $(1+p^c)$

$(03)\quad$ $\gcd\, (y,x)=1$, where $y={\rm lcm}\,(c,b)$ .

**Question.** Is there an example of such integers $x,p,q,y$?

**Note**:I edited the question to show why I posted this question and to make it clear

Thank you for any help .

p,q,xbe such that (1),(2),(3); then gcd(x,y) = 1.” But then you ask for an example of such integers. Do you mean acounterexampleto the implication? Or an example with (1),(2),(3)andwith gcd(y,x) = 1? This needs to be clarified, or the question is unanswerable. $\endgroup$ – Peter LeFanu Lumsdaine Jan 27 '16 at 11:09