Problem #292: Let $A$ be the set of $n\in \mathbb{N}$ such that there...

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Progress

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.