On $t$-sumfree sequences

Daan van Berkel; Wieb Bosma

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arXiv · 2609.16843

On $t$-sumfree sequences

Daan van Berkel·Wieb Bosma

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Abstract

In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$. Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values. The main open problem concerns the question of the conjectured ultimate periodicity of these sequences. We formulate a sufficient condition to prove that a conjectured preperiod and period are correct. We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences. Finally, we present the results of extensive computations on 3-sumfree sequences.

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Daan van Berkel, Wieb Bosma. 2026-09-15. On $t$-sumfree sequences. https://arxiv.org/abs/2609.16843

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