Experimental Mathematics

Iterating the sum-of-divisors function

Graeme L. Cohen and Herman J. J. te Riele


Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any fixed $m\ge1$, are there infinitely many $(m,k)$-perfect numbers? Is every positive integer $(m,k)$-perfect, for sufficiently large $m\ge1$? In this connection, we have obtained the smallest value of $m$ such that $n$ is $(m,k)$-perfect, for $1\le n\le1000$. We also address questions concerning the limiting behaviour of $\sigma^{m+1}(n)/\sigma^m(n)$ and $(\sigma^m(n))^{1/m}$, as $m\to\infty$.

Article information

Experiment. Math., Volume 5, Issue 2 (1996), 91-100.

First available in Project Euclid: 13 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11Y55: Calculation of integer sequences


Cohen, Graeme L.; te Riele, Herman J. J. Iterating the sum-of-divisors function. Experiment. Math. 5 (1996), no. 2, 91--100. https://projecteuclid.org/euclid.em/1047565640

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  • See authors' correction: Errata: Iterating the sum-of-divisors function. Experiment. Math. vol. 6, iss. 2 (1997), p. 177.