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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 337.10005

**Autor: ** Erdös, Paul

**Title: ** On asymptotic properties of aliquot sequences. (In English)

**Source: ** Math. Comput. 30, 641-645 (1976).

**Review: ** If n is a positive integer the aliquot sequence **{**s^{i}(n) **}** with leader n is defined as follows: s^{o}(n) = n and s^{k+1}(n) = \sigma (s^{k}(n))-s^{k}(n) for k \geq 0. The Catalan-Dickson conjecture states that every aliquot sequence is bounded (so that either s^{k}(n) = 1 for some k or the sequence becomes periodic). Guy and Selfridge, however, are ``tempted to conjecture'' that the Catalan-Dickson conjecture is false. The main result of the present paper is as follows: for every positive integer k and every positive real number \delta (1- \delta)n(s(n)/n)^{i} < s^{i}(n) < (1+\delta)n(s(n)/n)^{i}, 1 \leq i \leq k (*) for all n except a sequence of density zero. Since s(n)/n \geq 7/5 for n \equiv 0(mod 30), (*) implies that for every k there exists an m such that s^{o}(m) < s(m) < s^{2}(m) < ... < s^{k}(m). The result just stated was first proved by H. W. Lenstra, and his proof is published for the first time in the present paper.

**Reviewer: ** P.Hagis jun

**Classif.: ** * 11B37 Recurrences

11A25 Arithmetic functions, etc.

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag