An upper bound for Erdösreciprocal sum of the sum-free sequence

Let A={<italic>a</italic>₁, <italic>a</italic>₁,…} be a strictly increasing sequence of positive integers. <italic>A</italic> is called a sum-free sequence if no <italic>a</italic>_i is the sum of two or more distinct earlier terms. By improving the properties of sum-free sequence, we de ne three types of new sequences, i.e., <italic>κ</italic>₁ sequence, <italic>κ</italic>₂₁ sequence and <italic>κ</italic>₂₂ sequence. With the aid of <italic>κ</italic>₁; <italic>κ</italic>₂₁; <italic>κ</italic>₂₂ sequences, by using the ideas of Erdös, Levine and O'Sullivan, and the idea and method of Chen, we improve the upper bounds of Erdösreciprocal sum constants. Let <italic>ρ</italic>(<italic>A</italic>)=Σ_{<italic>k</italic>=1}^∞ 1/<italic>a</italic>_{<italic>k</italic>}. We improve the upper bound of <italic>ρ</italic>(<italic>A</italic>) to 2.86 or less, and prove that <italic>ρ</italic>(<italic>A</italic>)&lt; 2.8570.