Problem Statement
Let $p(x)\in \mathbb{Q}[x]$. Is it true that\[A=\{ p(n)+1/n : n\in \mathbb{N}\}\]is strongly complete, in the sense that, for any finite set $B$,\[\left\{\sum_{n\in X}n : X\subseteq A\backslash B\textrm{ finite }\right\}\]contains all sufficiently large integers?
Categories:
Number Theory Complete Sequences
Progress
Graham [Gr63] proved this is true when $p(n)=n$. Erdős and Graham also ask which rational functions $r(x)\in\mathbb{Z}(x)$ force $\{ r(n) : n\in\mathbb{N}\}$ to be complete?Graham [Gr64f] gave a complete characterisation of which polynomials $r\in \mathbb{R}[x]$ are such that $\{ r(n) : n\in \mathbb{N}\}$ is complete.
In the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from [Gr63] together with result of Alekseyev [Al19] mentioned in [283].
Source: erdosproblems.com/351 | Last verified: January 14, 2026