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/ai < 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+x1-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)/x1-\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

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