On the reciprocal sum of a sum-free sequence

Abstract

Let A = {1 ⩾ a ₁ < a ₂ < …} be a sequence of integers. A is called a sum-free sequence if no a _i is the sum of two or more distinct earlier terms. Let λ be the supremum of reciprocal sums of sum-free sequences. In 1962, Erdős proved that λ < 103. A sum-free sequence must satisfy a _n ⩽ (k + 1)(n − a _k ) for all k, n ⩽ 1. A sequence satisfying this inequality is called a κ-sequence. In 1977, Levine and O’sullivan proved that a κ-sequence A with a large reciprocal sum must have a ₁ = 1, a ₂ = 2, and a ₃ = 4. This can be used to prove that λ < 4. In this paper, it is proved that a κ-sequence A with a large reciprocal sum must have its initial 16 terms: 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ < 3.0752. Three conjectures are posed.