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Claim. Y.-G. Chen and W.-X. Yu, On $d$-complete sequences of integers, II, Acta Arith. 207 (2023), 161--181 (online 20 March 2023). As the paper's abstract and its zbMATH summary (Zbl 1528.11016) state, the authors give a criterion for the $d$-completeness of $\{p^aq^br^c\}$ for pairwise coprime $p,q,r\ge2$ and apply it to prove that $\{3^a5^br^c\}$ is $d$-complete for $1<r\le14$ with $(r,15)=1$, $\{2^a5^br^c\}$ for $1<r\le87$ with $(r,10)=1$, and $\{2^a7^br^c\}$ for $1<r\le33$ with $(r,14)=1$; they also answer how sparse a $d$-complete sequence can be. For Problem 123 this settles every triple in those three ranges.

Covers. The triples $(2,5,r)$ for $1<r\le87$, $(2,7,r)$ for $1<r\le33$ and $(3,5,r)$ for $1<r\le14$, each $r$ coprime to the other two bases. Not covered: every other triple.

Depends on. Nothing in this wiki; the result rests on the cited paper. It extends Erdős and Lewin's triples and Ma and Chen's.

Read depth. The statement is taken from the paper's abstract on the publisher's page and its zbMATH summary; the body of the paper, which the library does not hold, was not read.

Acceptance. Refereed: Acta Arithmetica 207 (2023). The site's commentary lists the result, but the site credits the settlement of the problem to Snyder's proof, so the commentary is not reviewed evidence for this paper.