Problem 460

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claims/: The 2 claim pages of Problem 460, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq i<k$. Does

$$\sum_{0<a_i< n}\frac{1}{a_i}\to \infty$$

as $n\to \infty$? What about if we restrict the sum to those $i$ such that $n-a_j$ is divisible by some prime $\leq a_j$, or the complement of such $i$?

Formulation. The site's wording, last edited 14 January 2026, follows [Er77c, p. 64]: $a_0=0$ and coprimality over $0\le i<k$, which forces $(a_k,n)=1$. [ErGr80, p. 91] takes $a_0=n$, $a_1=1$ and $1\le i<k$, and the site stated that form until its edit of 14 January 2026. Neither source restricts the sum to $a_i<n$. Without that restriction the sum is infinite, since $a=n+p$ is chosen for every prime $p>n$. This holds for every $n\ge2$ in the site's formulation and every $n\ge3$ in the monograph's, whose sequence ends after finitely many terms when $n=2$. The site credits the observation to the account Svyable using ChatGPT, and the same argument appears in the January 2026 notes of Benson and Chojecki. Sven Hardy Benson's manuscript A Robust Solution to Erdős Problem #460 (January 2026, PDF, linked from the site's discussion thread on 14 January 2026) proves, in the monograph's form, that the uncut sum diverges for every $n\ge3$, that its subseries over the indices $i$ for which some prime $\le a_i$ divides $n-a_i$ diverges, and that the complementary subseries is finite; it answers only this uncut wording, so it has no claim page. The site added the restriction $0<a_i<n$ and marks the original source as ambiguous, and the community database marks the statement ambiguous. The page shows only the site's Statement, so the standing answers the site's wording.

Status. Open, the site's label (OPEN; page last edited 14 January 2026). Przemyslaw Chojecki's notes of 13 and 14 January 2026 claimed the divergence of the truncated sum in both formulations; after the site's curator replied that they do not establish $f(n)\to\infty$ for every $n$, the author called the result a reduction, and the claim is recorded as withdrawn. The part the notes prove, the lower bound $S_{<n}(n)\ge f(n)$ with $\limsup_{n\to\infty}S_{<n}(n)=\infty$, is a pending partial claim. Claim pages: Chojecki, 13 January 2026 (withdrawn) and Chojecki, 14 January 2026 (claimed, partial).

Source. erdosproblems.com/460, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #460, https://www.erdosproblems.com/460.

References.

• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Formalization. None recorded.

Current assessment

The question as the site states it is open in all three forms: the truncated sum $S_{<n}(n)=\sum_{0<a_i<n}1/a_i$ and its two restricted sums. The site's commentary records that the question arose in work of Eggleton, Erdős and Selfridge, who proved $a_k<k^{2+o(1)}$ for $k$ large in terms of $n$ and conjectured $a_k\ll k\log k$; both sources announce a forthcoming paper of theirs, which the site's curator could not find, so the intended problem is uncertain, as the Formulation records.

Write $P^-(m)$ for the least prime factor of $|m|$ and

$$f(n)=\sum_{\substack{1\le a<n\\ P^-(n-a)>a}}\frac1a.$$

Chojecki's notes prove that every $a<n$ with $P^-(n-a)>a$ is a term of the sequence, in both formulations, because such an $a$ is admissible as soon as it becomes available; hence $S_{<n}(n)\ge f(n)$. They prove $\frac1N\sum_{n\le N}f(n)\gg\log\log N$ from Buchstab's asymptotic for rough numbers, hence $\limsup_{n\to\infty}S_{<n}(n)=\infty$. The site's commentary credits Chojecki with the reduction to $f(n)\to\infty$ and attributes the averaged bound to standard estimates on rough numbers. Whether $f(n)\to\infty$, and so whether $S_{<n}(n)\to\infty$, is open; the revised note's Conjecture 13, a multiscale lower bound for rough numbers near $n$, would give it. The first note identifies the terms $a_k<n$ with $P^-(n-a_k)>a_k$ as exactly the integers counted by $f(n)$, so the truncated sum over the complementary indices of the statement, those $i$ for which no prime $\le a_i$ divides $n-a_i$, equals $f(n)$; nothing beyond the averaged bound is recorded for either restricted sum. The author reports using GPT-5.2 for most of the work.

No formalization is recorded. A thread comment of 15 January 2026 reports two Lean statements proved and one negated by Aristotle, an automated prover: the divergence of the prime reciprocal sum and of its tails, and a negated existence statement for the sequence, posted without a file; it is not a formalization of the problem. No search beyond the site, its discussion thread and the notes linked there is recorded here.