Iterating the Sum-of-Divisors Function

Herman J. J. te Riele

Abstract

Let σ^{0}(n) = n and σ^{m}(n) = σ(σ^{m−1}(n)), where m ≥ 1 and σ is the sum-of-divisors function. We say that n is (m, k)-perfect if σ^{m}(n) = kn. We have tabulated all (2, k)-perfect numbers up to 10^{9} and all (3, k)- and (4, k)-perfect numbers up to 2·10^{8}. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any fixed m ≥ 1, are there infinitely many (m, k)- perfect numbers? Is every positive integer (m, k)-perfect, for sufficiently large m ≥ 1? In this connection, we have obtained the smallest value of m such that n is (m, k)-perfect, for 1 ≤ n ≤ 1000. We also address questions concerning the limiting behaviour of σ^{m+1}(n)/σ^{m}(n) and (σ^{m}(n))^{1/m}, as m → ∞.