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 = {ai} is of positive upper logarithmic density, i. e. if

(^*) limsupx ––> oo (log x)-1 sumai < x ai-1 > 0

holds, then there exists an infinite increasing subsequence {anj} \subset A, satisfying ani | anj. Such a subsequence is called a chain. Using the notations log2 x = log log x and limsup for limsupx ––> oo, and with ci > 0, the following sharper results are now proved; (1) If A satisfies (^*), then it contains a chain, satisfying for infinitely many y: sumai < y 1 > c1(log2 y) ½. (2) If satisfies limsup (log2 x)-1 suman < x (an log an)-1 = c2 then it contains a chain, satisfying for infinitely many x: suma_{ni < x} 1 > c3 log2 x. It is shown that (1) is best possible, by exhibiting a class of sequences, for which all chains are such that suma_{ni < x} 1 < 3(log2 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 (log2 y)-1 suma_{ni < y} \geq limsup (log2 x)-1 suman < x (an log an)-1.

(b) For every sequence A,

limsup (log x)-1 sumai < x,ak | ai (ak /ai) \geq limsup (log x)-1 sumak \leq x ak-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

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