A-Sequence

An infinite sequence of positive integers satisfying

(1)

is an -sequence if no is the sum of two or more distinct earlier terms (Guy 1994). Such sequences are sometimes also known as sum-free sets.

Erdős (1962) proved

(2)

Any -sequence satisfies the chi inequality (Levine and O'Sullivan 1977), which gives . Abbott (1987) and Zhang (1992) have given a bound from below, so the best result to date is

(3)

Levine and O'Sullivan (1977) conjectured that the sum of reciprocals of an -sequence satisfies

(4)

where are given by the Levine-O'Sullivan greedy algorithm. However, summing the first terms of the Levine-O'Sullivan sequence already gives 3.0254....