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Claim. Theorem 1 of G. Melfi, On the conditional infiniteness of primitive weird numbers (p. 509; Melfi 2015): let $k\ge1$ and let $a$, $b$ be positive odd integers such that $p=2^{k+2}-a$ and $q=2^{k+2}+b$ are prime. If $b+3<a<2^{(k-1)/2}$, then $n=2^kpq$ is a primitive weird number. The paper deduces (pp. 509--510) that if $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for all sufficiently large $n$, where $p_n$ is the $n$th prime, then there are infinitely many primitive weird numbers, all of the form $2^kpq$. Under that hypothesis this answers the second question of Problem 470 yes. The first question, on odd weird numbers, is not answered: the paper remarks only (Section 4) that the proof of Theorem 1 extends with $2^k$ replaced by an almost perfect number $m$, one with $\sigma(m)=2m-1$, so that an odd almost perfect number greater than $1$ would give an odd weird number. The paper takes weird to mean $\sigma(n)>2n$ and not a sum of distinct proper divisors of $n$, the reading recorded under the problem page's Formulation.

Hypothesis. The prime-gap bound $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for all large $n$ is unproved. Cramér's conjecture $p_{n+1}-p_n=O((\log p_n)^2)$ would give it, and so would the much weaker conjecture, which the paper attributes to Gonek, that $p_{n+1}-p_n<p_n^{\varepsilon}$ for every $\varepsilon>0$ and all large $n$. The unconditional bound $p_{n+1}-p_n<p_n^{0.525}$ of Baker, Harman and Pintz, which the paper calls very close to what is needed, is not enough. The claim gives no unconditional answer.

Depends on. Nothing in this wiki; the hypothesis is stated above.

Acceptance. Published in J. Number Theory 147 (2015), 508--514, a refereed journal, DOI 10.1016/j.jnt.2014.07.024 (refereed); the article was available online on 16 September 2014, which dates this page. No arXiv posting is recorded. The site credits the conditional result in its commentary but labels the problem OPEN, so that credit is not acceptance and no reviewed evidence is listed.