Counting Egyptian fractions

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$\frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N,$$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.