On -complete sequences of integers, II
Volume 207 / 2023
Abstract
In 1996, Erdős and Lewin introduced the notion of d-complete sequences. A sequence \mathcal T of positive integers is called d-complete if every sufficiently large integer can be represented as the sum of distinct terms taken from \mathcal T such that no one divides any other. It is known that for any positive integers q \gt p \gt 1, the sequence \{ p^aq^b : a, b=0,1,\dots \} is d-complete if and only if \{ p, q \} =\{ 2, 3\} . Let p,q,r be three pairwise coprime integers not less than 2. In this paper, we establish a criterion for the d-completeness of the general sequence \{ p^a q^b r^c : a, b, c=0,1,\dots \}. As applications, we extend earlier results and prove that \{ 3^a5^br^c : a, b, c=0,1,\dots \} is d-complete for 1 \lt r\le 14 with (r, 15)=1, \{ 2^a 5^b r^c : a, b, c=0,1,\dots \} is d-complete for 1 \lt r\le 87 with (r, 10)=1 and \{ 2^a7^br^c : a, b, c=0,1,\dots \} is d-complete for 1 \lt r\le 33 with (r, 14)=1. We also give an answer to the following question: how sparse can a d-complete sequence be? Moreover, we pose a problem for further research.