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 $y=11x$?

[Wolfram|Alpha calculation]:http://www.wolframalpha.com/input/?i=Table+of[sigma_11%28n%29+mod+sigma_1%28n%29]%2C+for+n%3D2+to+1000