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**Zentralblatt MATH**

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

**Zbl.No: ** 146.27102

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

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

**Source: ** Acta Arith. 11, 411-418 (1966).

**Review: ** Let A = **{**a_{n}**}** be a sequence of integers; set f(x) = **sum**_{ai | aj,aj \leq x} 1. The main result of this paper is Theorem 1. If A has positive upper logarithmic density c_{1}, then there exists c_{2}, depending on c_{1} only, so that for infinitely many x, f(x) > x\exp**{**c_{2}(log_{2} x)^{ ½} log_{3} x**}**. On the other hand, there exists a sequence A of positive upper logarithmic density c_{1}, so that for all x,f(x) < x\exp**{**c_{3}(log_{2} x)^{ ½} log_{3} x**}**. (All c_{i} stand for positive constants and log_{k}x for the iterated logarithm.) The first inequality is proved using a purely combinatorial Theorem: Let *S* be a set of n elements and let B_{1},...,B_{2}, z > c_{4}2^{n} (0 < c_{4} < 1) be subsets of *S*. Then, if n > n_{0}(c_{4}), one of B's contains at least \exp(c_{5} n^{ ½} log n) of the B's, where c_{5} depends only on c_{4}. The second inequality is proved using a result of probabilistic number theory.

**Reviewer: ** E.Grosswald

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

11B05 Topology etc. of sets of numbers

**Index Words: ** number theory

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