Problem #309: Let $N\geq 1$. How many integers can be written as the sum...

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Progress

If the number of such integers is $N(n)$ then it is trivial that $N(n)\leq \log n+O(1)$. Yokota [Yo97] proved that $N(n)\geq \log n-O(\log\log n)$.

Croot [Cr99] proved that every integer at most\[\leq \sum_{n\leq N}\frac{1}{n}-(\tfrac{9}{2}+o(1))\frac{(\log\log N)^2}{\log N}\]can be so represented.

If $F(N)$ counts the number of integers which can be represented in this fashion, then the current best lower bound known is\[F(N) \geq \log N+\gamma -\left(\frac{\pi^2}{3}+o(1)\right)\frac{(\log\log N)^2}{\log N}\]due to Yokota [Yo02].