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\$.

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CWI
Department of Numerical Mathematics [NM]
Large-scale computing

Cohen, G.L, & te Riele, H.J.J. (1995). Iterating the sum-of-divisors function. Department of Numerical Mathematics [NM]. CWI.