Compute $F(k)$, the number of representations of $1$ as a sum of $k$ distinct unit fractions (Erdős #148)

For $k\ge 1$ let $F(k)$ be the number of solutions in integers to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with $1\le n_1<n_2<\cdots<n_k$ (all $n_i$ distinct). Erdős and Graham asked for good estimates of $F(k)$. For any fixed $k$ this count is a FINITE enumeration: the greedy/harmonic bound forces $n_i\le k\cdot 2^{2^{k}}$-type limits and standard denominator bounds prune the search, so $F(k)$ can be computed exactly by backtracking. GOAL: extend the exact sequence of values $F(k)$ (OEIS A006585) to the largest $k$ feasible, and compare the empirical growth to the known asymptotic bounds.

Acceptance. PARTIAL / PRIMARY: compute $F(k)$ exactly for one or more values of $k$ beyond the largest currently tabulated term of OEIS A006585, delivered as (i) the exact integer value(s), (ii) a runnable enumeration script, and (iii) a cross-check that it reproduces all known terms of A006585 on the overlap. A stronger partial: fit the computed $\log F(k)$ against the Konyagin lower and Elsholtz–Planitzer upper bounds and report which is closer to the data, with the ratios. FULLY RESOLVES (analytic, not finite): a proof narrowing the asymptotic gap, e.g. establishing $\log F(k)$ has a determined growth order. State the search bound on the $n_i$ used and argue its completeness.