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Claim. P. Erdős and M. Lewin, $d$-complete sequences of integers, Math. Comp. 65 (1996), no. 214, 837--840. Call a sequence $d$-complete when every sufficiently large integer is a sum of distinct terms of it, no one of which divides another. Theorem 2 (p. 839): the sequence $\{2^\alpha5^\beta p^\gamma\}$ is $d$-complete for every prime $p$ with $6<p<20$. Proposition 4 (p. 839): the sequence $\{3^\alpha5^\beta7^\gamma\}$ is $d$-complete, every integer above $185$ being representable. Alongside Theorem 2 the paper gives the largest integers not so representable, $31$, $34$, $24$, $115$ and $155$ for $p=7,11,13,17,19$, with its Propositions 2 and 3 for $p=11$ and $p=19$; the paper notes that its method meets difficulty at $p=23$. These are the first settled triples of Problem 123, whose question is the paper's conjecture (i) on p. 840. Library home: erdos_1996_d_complete_sequences_integers.

Covers. The triples $(2,5,p)$ for $p\in\{7,11,13,17,19\}$ and the triple $(3,5,7)$. Not covered: every other pairwise coprime triple.

Depends on. Nothing in this wiki; the result rests on the cited paper.

Read depth. The statements were checked against the paper on its library card; the proofs were read there but not verified, and Propositions 2 to 4 rest partly on inspections that the paper reports without listing and that were not repeated.

Acceptance. Refereed: Mathematics of Computation 65 (1996), no. 214. The site's commentary lists these cases, but the site credits the settlement of the problem to Snyder's proof, so the commentary is not reviewed evidence for this paper. Later work extended the triples: Ma and Chen and Chen and Yu.