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1996 Iterating the sum-of-divisors function
Graeme L. Cohen, Herman J. J. te Riele
Experiment. Math. 5(2): 91-100 (1996).

Abstract

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 $m\to\infty$.

Citation

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Graeme L. Cohen. Herman J. J. te Riele. "Iterating the sum-of-divisors function." Experiment. Math. 5 (2) 91 - 100, 1996.

Information

Published: 1996
First available in Project Euclid: 13 March 2003

zbMATH: 0866.11003
MathSciNet: MR1418956

Subjects:
Primary: 11A25
Secondary: 11Y55

Rights: Copyright © 1996 A K Peters, Ltd.

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Vol.5 • No. 2 • 1996
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