##
**Zentralblatt MATH**

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

**Zbl.No: ** 146.27103

**Autor: ** Erdös, Pál; Sarközi, A.; Szemeredi, E.

**Title: ** On divisibility properties of sequences of integers (In English)

**Source: ** Stud. Sci. Math. Hungar. 1, 431-435 (1966).

**Review: ** *H.Davenport* and *P.Erdös* (Zbl 015.10001) proved that if the sequence A = **{**a_{i}**}** is of positive upper logarithmic density, i. e. if (^*) **limsup**_{x ––> oo} (log x)^{-1} **sum**_{ai < x} a_{i}^{-1} > 0 holds, then there exists an infinite increasing subsequence **{**a_{nj}**}** \subset A, satisfying a_{ni} | a_{nj}. Such a subsequence is called a chain. Using the notations log_{2} x = log log x and **limsup** for **limsup**_{x ––> oo}, and with c_{i} > 0, the following sharper results are now proved; (1) If A satisfies (^*), then it contains a chain, satisfying for infinitely many y: **sum**_{ai < y} 1 > c_{1}(log_{2} y)^{ ½}. (2) If satisfies **limsup** (log_{2} x)^{-1} **sum**_{an < x} (a_{n} log a_{n})^{-1} = c_{2} then it contains a chain, satisfying for infinitely many x: **sum**_{a_{ni} < x} 1 > c_{3} log_{2} x. It is shown that (1) is best possible, by exhibiting a class of sequences, for which all chains are such that **sum**_{a_{ni} < x} 1 < 3(log_{2} x)^{ ½}; and also, that, in general, (2) will not hold for all x. The following two conjectures are stated: (a) For every sequence A, there is a chain satisfying

**limsup** (log_{2} y)^{-1} **sum**_{a_{ni} < y} \geq **limsup** (log_{2} x)^{-1} **sum**_{an < x} (a_{n} log a_{n})^{-1}. (b) For every sequence A,

**limsup** (log x)^{-1} **sum**_{ai < x,ak | ai} (a_{k} /a_{i}) \geq **limsup** (log x)^{-1} **sum**_{ak \leq x} a_{k}^{-1}.

(Remark: On the first line the word "lower" sems to stand for "upper".)

**Reviewer: ** E.Grosswald

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B05 Topology etc. of sets of numbers

**Index Words: ** number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag