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Claim. The answer to Problem 281 is yes, by two classical theorems. Write $B_k$ for the multiples of $n_1,\ldots,n_k$, a periodic set whose density we call $\delta_k$, and $B$ for the multiples of the whole sequence. Davenport and Erdős (Acta Arith. 1936) prove that the lower natural density of $B$ equals $\lim_k\delta_k$; the library card records that theorem, Theorem 1(b) of the paper, and this application. Under the problem's hypothesis with every $a_i=0$, the integers outside $B$ have density $0$, so $B$ has density $1$ and $\delta_k\to1$: for every $\epsilon>0$ there is $k$ with $1-\delta_k<\epsilon$. Rogers' theorem, which Halberstam and Roth's Sequences (1966, Chapter V.3) first published, states that for fixed moduli $n_1,\ldots,n_k$ the density of the integers in none of the classes $a_i\pmod{n_i}$ is largest when every $a_i=0$. So for every residue choice the integers avoiding the first $k$ classes have density at most $1-\delta_k<\epsilon$. The argument uses the hypothesis only for the zero residues, so it proves more than asked: it needs only that the integers divisible by none of the $n_i$ have upper density $0$.

As first posted (2026-01-18), the observation cited Theorem 12 of Chapter V of Halberstam and Roth, that the logarithmic density of $B$ equals $\lim_k\delta_k$, and Rogers' theorem from p. 242 of that book; Terence Tao traced the density statement to the 1936 paper of Davenport and Erdős the same day, and the site's commentary gives the argument in that form. R. R. Hall and G. Tenenbaum's paper On Behrend sequences (Math. Proc. Cambridge Philos. Soc. 112 (1992), 467--482), linked from a follow-up comment, spells out the zero-residue step: by the Davenport-Erdős theorem, when almost all integers are multiples of the sequence, the multiples of a long enough initial segment have density at least $1-\epsilon$. The paper does not treat other residues or Rogers' theorem. Tao posted a proof of Rogers' theorem on Tao's blog on 2026-01-19, since the theorem has no separate published source. The thread remarks that the extension of the Davenport-Erdős theorem to nonzero residues, which Erdős asked about, is Problem 25, generalized by Problem 486.

Depends on. Nothing in this wiki. The proof is independent of Somani's argument, which settles the problem by a different route.

Acceptance. Reviewed: the site's curator, Thomas Bloom, wrote this argument into the problem's commentary as an alternative elementary proof and credits KoishiChan in the page's acknowledgments (page last edited 18 January 2026). On the thread, Terence Tao confirmed on 2026-01-18 that the result follows from the Davenport-Erdős theorem once Rogers' theorem is applied, and reported, quoting with permission, that Gérald Tenenbaum confirmed by email that the solution is immediate given the two classical results. Not refereed: the two ingredients are published theorems, and their combination is a forum observation. No formalization of this route is known to this corpus; the Lean development recorded on Somani's page formalizes the other argument.