##
**Zentralblatt MATH**

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

**Zbl.No: ** 147.02601

**Autor: ** Erdös, Pál

**Title: ** On the difference of consecutive terms of sequences defined by divisibility properties (In English)

**Source: ** Acta Arith. 12, 175-182 (1966).

**Review: ** Let A denote a set of pairwise relatively prime positive integers such that **sum** 1/a_{i} < oo. Let B be the set of positive integers not divisible by any element of A. The author proves: I. There is an absolute constant c > 0 (independent of A) such that for sufficiently large x the interval (x,x+x^{1-c}) contains ab in B. II. There is a set A such that for infinitely many b and all b' > b(b,b' in B) we have b'-b > \exp **(** ^{1}/_{4} (log b log log b)^{ ½}**)**. III. Let \beta be the density of B and let f(x)/x^{1-\epsilon} ––> oo for all \epsilon > 0 then B(x,x+f(x)) = (\beta+o(1))f(x). (Here B(u,v) is the number of b in B such that u < b < v).

**Reviewer: ** H.B.Mann

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

**Index Words: ** number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag