Note on sequences of integers no one of which is divisible by any other. (English) Zbl 0012.05202

It was proved recently by A.S. Besicovitch [Math. Ann. 110, 336–341 (1934; Zbl 0009.39504)] that a sequence \(a_1,a_2, \dots\) 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 F. Behrend see the following review of [J. Lond. Math. Soc. 10, 42–44 (1935; Zbl 0012.05203)].) In the above connection Besicovitch (loc. cit.) proved that if \(d_a\) denotes the density of those integers which have a divisor between \(a\) and \(2a\), then \(\lim_{a \to \infty} \inf d_a=0\). It is shown 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 \(a\).


11B83 Special sequences and polynomials
11N25 Distribution of integers with specified multiplicative constraints
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