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Claim. The answer to Problem 26 is no: there is an infinite set $A\subset\mathbb{N}$ such that for no $k\geq 1$ do almost all integers have a divisor of the form $a+k$ with $a\in A$.

The theorem. Let $a_1<a_2<\cdots$ be a sequence of positive integers and let $A_m$ be the density of the integers divisible by at least one of $a_1,\ldots,a_m$. Davenport and Erdős proved that the set of all multiples of the sequence has lower density and logarithmic density both equal to $\lim_m A_m$, and, as the easy case they isolate first, that when $\sum_j 1/a_j$ converges the set of multiples has an ordinary density equal to that limit (pp. 19--23 of the paper; the card Davenport and Erdős 1951 digests it). The paper does not state that the limit is then below one; that step is this page's own and needs every $a_j\geq 2$ (with $a_1=1$ every integer is a multiple): the integers coprime to $a_1\cdots a_m$ have positive density and none is a multiple of $a_1,\ldots,a_m$, while for $j>m$ the multiples of $a_j$ among them make up at most a fraction $1/a_j$ of them, so once $\sum_{j>m}1/a_j<1$ a positive fraction of that set has no divisor in the sequence at all. Tenenbaum's survey (Tenenbaum 2013) reaches the general statement from the Davenport--Erdős formula for the lower density of a set of multiples as the limit over its finite stages (his (25)) and Behrend's inequality (his (26)): a divergent reciprocal sum is necessary for a Behrend sequence (his (27)).

The disproof. Call a set Behrend when almost all integers have a divisor in it. By the theorem with the step above, or by Tenenbaum's (27), a Behrend set of integers $\geq 2$ has a divergent reciprocal sum. If $A$ is infinite with $\sum_{a\in A}1/a<\infty$, then for every $k\geq 1$ the set $A+k$ has every element at least $2$ and a convergent reciprocal sum as well, so no $A+k$ is Behrend. Any such $A$, the powers of two for instance, answers the question in the negative for every shift. The site's commentary credits Davenport and Erdős, with Tenenbaum's survey as a second reference, for the divergence of the reciprocal sum of every Behrend sequence, draws this consequence and notes that the theorem is forty years older than the question, which Erdős and Tenenbaum asked. An explicit construction with the same effect, and the formalization the site records, is on the claim page Ruzsa's counterexample.

Acceptance. The site's curator, T. F. Bloom, marks the problem disproved and credits the Davenport--Erdős theorem for the negative answer, which the page lists as reviewed. The paper is H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. (N.S.) 15 (1951), 19--24, a refereed journal, listed as refereed. The site's label carries a Lean qualification; the Lean file it refers to formalizes Ruzsa's construction as its own statement and, for the formal-conjectures variant, proves the convergent-sum disproof for the set $\{2^{2^n}\}$ by a direct union bound on the densities of the multiples, not through this theorem; this corpus has not built it, so no formalized evidence is listed. The page is dated by the publication year alone: the paper's record gives no month and no DOI, so the first day of 1951 stands in for the issue date, which the paper's own "Received January 31, 1951" line shows the stand-in precedes.