Iterating the sum-of-divisors function

We define $sigma^0 (n)=n$ and $sigma^m (n)=sigma( sigma^{m-1 (n))$ ($m ge 1$), where $sigma$ is the sum-of-divisors function, and we call $n$ $(m,k)$-perfect if $sigma^m (n)=kn$. All $(m,k)$-perfect numbers $n$ are tabulated, for $n< 10^9$ ($m=2$) and $n<2 cdot 10^8$ ($m=3$, 4). These suggest a number of new results, which we subsequently prove, and a number of conjectures. We ask in particular: (1) For any fixed $mge1$, are there infinitely many $(m,k)$-perfect numbers? (2) Is every $n$ $(m,k)$-perfect, for sufficiently large $mge1$? In this connection, we list the smallest value of $m$ such that $n$ is $(m,k)$-perfect, for $1 le n le 400$. We also address questions concerning the limiting behaviour of $sigma^{m+1 (n)/ sigma^m (n)$ and $(sigma^m (n))^{1/m$, as $mtoinfty$.