OPEN

Let $F(k)$ be the number of solutions to
\[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\]
where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.

The current best bounds known are
\[2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},\]
where $c>0$ is some absolute constant and $c_0=1.26408\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin

[Ko14] and the upper bound to Elsholtz and Planitzer

[ElPl21].