What is the largest $A\subseteq\{1,\dots,N\}$ such that all subset sums $\sum_{n\in S}1/n$ (over $S\subseteq A$) are distinct?

Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $$R(N) \le \frac{1}{\log 2} \log_r N \left( \frac{N}{\log N} \prod_{i=3}^{r} \log_i N \right),$$ valid for any $k \ge 4$ with $\log_k N \ge k$ and any $r \ge 1$ with $\log_{2r} N \ge 1$. (In these bounds $\log_i n$ denotes the $i$-fold iterated logarithm.)

[BlEr75] Bleicher, M. N. and Erdős, P., _The number of distinct subsums of Unparsed notation: \sum \sb{1}\spN\,1/i_. Math. Comp. (1975), 29-42. [BlEr76b] Bleicher, Michael N. and Erdős, Paul, _Denominators of Egyptian fractions. II_. Illinois J. Math. (1976), 598-613.