##
**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 a_{1},a_{2},... 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 a_{n} log a_{n}} < 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 d_{a} denotes the density of those integers which have a divisor between a and 2a, then **lim**_{a ––> oo} **inf** d_{a} = 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag