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**Zentralblatt MATH**

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

**Zbl.No: ** 015.10001

**Autor: ** Davenport, H.; Erdös, Pál

**Title: ** On sequences of positive integers. (In English)

**Source: ** Acta Arith. 2, 147-151 (1936).

**Review: ** Let a_{1},a_{2},... be any sequence of different positive integers, and b_{1},b_{2},... the integers divisible by at least on a. It was proved by *A.S.Besicovitch* (Zbl 009.39504) that the sequence **{**b_{i}**}** may have different upper and lower densities. Here it is shown that the logarithmic density **lim**_{x ––> oo} (log x)^{-1} **sum**_{bi \leq x} b_{i}^{-1} exists and is equal to the lower density of the sequence. The proof uses Dirichlet series. It is deduced that if a sequence a_{1},a_{2},... has a positive upper logarithmic density, then it has a subsequence a_{i1},a_{i2},... in which a_{ik} | a_{i_{k+1}} (k = 1,2,...).

**Reviewer: ** E.C.Titchmarsh (Oxford)

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

11B05 Topology etc. of sets of numbers

**Index Words: ** Algebra, number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag