Periodicity conjectures for all 2-sumfree sequences

Source abstract

Complementing an earlier paper, which focused on $3$-sumfree sequences, we here consider only $2$-sumfree sequences: starting with positive integers $f$ and $g>f$, the infinite, increasing 2-sumfree sequence $S_{f,g}$ is constructed as follows. After any initial segment, the next entry is the smallest positive integer exceeding all previous ones that differs from all sums of distinct pairs in the sequence. It follows from a theorem in the previous paper that for every $f\geq 1$ and all $f+1\leq g<2f$ the sequence $S_{f,g}$ exhibits ultimately periodic behaviour. In this paper we state precise conjectures that, if true, would imply that every $2$-sumfree sequence is ultimately periodic. Here periodicity of an increasing sequence is understood to mean periodicity of the sequence of first differences, or, equivalently, of its characteristic sequence. We supply much computational evidence to support the conjectures.