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 {si(n) } with leader n is defined as follows: so(n) = n and sk+1(n) = \sigma (sk(n))-sk(n) for k \geq 0. The Catalan-Dickson conjecture states that every aliquot sequence is bounded (so that either sk(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 < si(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 so(m) < s(m) < s2(m) < ... < sk(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.

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