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Claim. The answer to the last question of Problem 1189 is yes: for every odd prime $p$ the divisors of $2^{p-1}p$ greater than one form an irreducible covering set, so infinitely many $n$ have the property that their divisors above one form an irreducible covering set. Sun calls $n$ a covering number when the integers can be covered by classes with distinct moduli among the divisors of $n$ greater than one, and primitive when no proper divisor of $n$ is a covering number; Theorem 1.4 of the paper classifies the primitive covering numbers with at most two prime divisors as exactly the $2^{p-1}p$, and Corollary 1.3 answers Erdős's 1980 question by combining this primitivity with Simpson's theorem that a minimal cover with least common multiple $N=\prod p^{\alpha_p}$ has more than $\sum\alpha_p(p-1)$ classes, so that among the subsets of the divisor set only the whole set can be the moduli of a cover. The divisor set of $12$ is the case $p=3$, Erdős's own example. The source card is Sun 2007, which lists its theorems; the inequality used is on the card Simpson 1985.

Covers. The fourth question alone. The count of irreducible covering sets of size $k$, the extreme values of the largest modulus and the maximum reciprocal sum are not addressed; the pending full claims of Pickhardt and of Snyder answer those, and both reuse this theorem for the fourth.

Depends on. Simpson's inequality, which the paper combines with its own classification of primitive covering numbers.

Acceptance. Refereed: Z.-W. Sun, On covering numbers, Integers 7 (2007), no. 2, A33, 11 pp., also printed in the proceedings volume Combinatorial Number Theory (de Gruyter, 2007), 443--453, the DOI linked above, which is the citation the problem page carries as [Su07]; the paper was first posted to arXiv on 2006-01-01, which gives the page its date. Integers 7 (2007), no. 2, is the journal's proceedings issue for the Integers Conference 2005, whose preface describes the volume as refereed. No reviewed evidence is listed: the site's curator states in the problem page's commentary that Sun's theorem settles the final question, but the site labels the problem as a whole OPEN (page last edited 8 April 2026), so that remark is context and not acceptance, and the 2026 manuscript of Pickhardt importing the theorem as established is not a review. Nothing here rests on a review by this project.