Note on Sequences of Integers No One of Which is Divisible By Any Other

[Extracted from the Journal of the London Mathematical Society, Vol . 1.0 (1935) .] 1 Let at , a 2 , a3 , . . . be a sequence of integers, say (A), such that am is not a divisor of an unless m = n . Chowla, Davenport, and I proposed the question whether the density of every sequence (A) is zero . Besicovitch proved that this was not so by showing that, if da is the density of integers having a divisor between a and 2a, then lim infda = 0. a-*u We can easily prove that the upper density of any sequence (A) does not exceed 2 . In fact, (A) cannot contain n+1 elements a1 , a2 , . . ., an+1 at most equal to 2n. For, if a. = 21 bm, where bm is odd, and so has at most n different values, two of the b's must be equal . If these correspond to indices ml, m2 , clearly am, is divisible by a,,,, if m l > m2T . We prove now that the lower density of (A) is zero § . This follows from the