I would like to know more about divisibility among power-divisor functions. Put $\sigma_k(n) = \sum_{d \mid n} d^k$ for all positive integers $k$ and $n$.

My question here is : for which positive integers $x$ and $y$ do we have that $\sigma_x(n) $ divides $\sigma_y(n) $ for all $n$?

EDIT01 : For instance, this is true for $y = 11$ and $x = 1$, at least for $n$ from $2$ to $1000$ (see this Wolfram|Alpha calculation). Could it be that the solutions are always of the form $y=11x$?