Erdős Problem #292

Let \(A\) be the set of \(n\in \mathbb{N}\) such that there exist \(1\leq m_1<\cdots <m_k=n\) with \(\sum\tfrac{1}{m_i}=1\). Explore \(A\). In particular, does \(A\) have density \(1\)? Straus observed that \(A\) is closed under multiplication. Furthermore, it is easy to see that \(A\) does not contain any prime power. The answer is yes, as proved by Martin [Ma00], who in fact proved that if \(B=\mathbb{N}\backslash A\) then, for all large \(x\), \[\frac{\lvert B\cap [1,x]\rvert}{x}\asymp \frac{\log\log x}{\log x},\] and also gave an essentially complete description of \(B\) as those integers which are small multiples of prime powers. van Doorn has observed that if \(n\in A\) (with \(n>1\)) then \(2n\in A\) also, since if \(\sum \frac{1}{m_i}=1\) then \(\frac{1}{2}+\sum\frac{1}{2m_i}=1\) also.