## On $d$-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.