Erdős Problem #346

declared status 'open'. Formalized: yes. Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that {UL} {LI} $A\backslash B$ is complete for any finite subset $B$ and {/LI} {LI} $A\backslash B$ is not complete for any infinite subset $B$.{/LI} {/UL} (Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.) Is it true that if $a_{n+1}/a_n \geq 1+\epsilon$ for some $\epsilon>0$ and all $n$ then$$\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?$$ Prize: no. Tags: complete sequences, number theory.

Notation is rendered from the stored source. The pinned checkout remains the exact record.