A lower bound for the number of Egyptian fractions

An \emph{Egyptian fraction} is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_r$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $\EE_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds \[ \frac{\ln\card{\EE_N}}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , \] where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erd\H{o}s [Illinois J. Math. 20 (1976), pp. 598--613] who established a similar bound but under the more stringent condition $\ln_k N\geq k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $\card{\EE_N}$ for large positive integers $N$, and we give a table of $\card{\EE_N}$ for $N \leq 154$.