A lower bound for the number of Egyptian fractions

An Egyptian fraction is a sum of the form 1/n₁ + ⋯ + 1/nᵣ where n₁, , nₖ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set EN of rational numbers that can be represented by Egyptian fractions with denominators not exceeding N. More precisely, we show that for every integer k ≥ 4 such that lnₖ N ≥ 3/2 it holds ln(|EN|)/ln 2 ≥ (2 - 3/lnₖ N)N/ln N∏ⱼ₌₃ᵏ lnⱼ N , where lnₖ denotes the k-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition lnₖ N≥ k and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of |EN| for large positive integers N, and we give a table of |EN| for N up to $154$.