I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$?

where :$\phi_{k}$ is iterating Euler - totient function and $\sigma_{k}$ is iterating sum divisor .

**Note** :I tried to evaluate the recent limit I accrossed this problem :can I  write  :$${\phi_{k}(({2}^{m-1})({2^m-1}))}=({2}^{m-1-k})\phi_{k}({2}^{m-1})$$ ? I know only that is true iff gcd $({2}^{m-1},{2^m-1})=1$ for $m\geq 1$ and $k=1$ ?


Thank you for any help