Problem 460

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 $i$ ?

Source: erdosproblems.com/460

Number theory

Wiki page

Status

Open

No claim settles this problem.

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do not establish $f(n)\to\infty$ for every $n$ , the author called the result a reduction, and the claim is recorded as withdrawn.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 Claim pages: Chojecki, 13 January 2026 (withdrawn) and Chojecki, 14 January 2026 (claimed, partial).

Problem 460

Status

Proof claims

Discussion