Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  012.05202
Autor:  Erdös, Paul
Title:  Note on sequences of integers no one of which is divisible by any other. (In English)
Source:  J. London Math. Soc. 10, 126-128 (1935).
Review:  It was proved recently by A.S.Besicovitch (Zbl 009.39504) that a sequence a1,a2,... of integers no one of which is divisible by any other does not necessarily have density zero. It is here proved that for such a sequence, sum {1\over an log an} < c, an absolute constant, so that the lower density is necessarily zero. (For a different proof by Behrend see the foll. review.) In the above connection Besicovitch (l.c.) proved that if da denotes the density of those integers which have a divisor between a and 2a, then lima ––> oo inf da = 0. It is shewn here that liminf may be replaced by lim. The proof follows easily from a result of the Hardy-Ramanujan type, which is roughly: the normal number of prime factors less than a of an integer is log log a for large a.
Reviewer:  Davenport (Cambridge)
Classif.:  * 11B83 Special sequences of integers and polynomials
                   11N25 Distribution of integers with specified multiplicative constraints
Index Words:  Algebra, number theory

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