PreprintCC BY-NC-ND 4.0arXiv
On t-sumfree sequences
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.
Related
- Periodicity conjectures for all 2-sumfree sequences
- Return-Growth Sequences: Euler Enumeration and Pattern Avoidance
- Adams Spectral Sequences and Franke’s Algebraicity Conjecture
- Design-Based Confidence Sequences: A General Approach to Risk Mitigation in Panel Experiments
- Spectral sequences, décalage, and the Beilinson t -structure
BibTeX
@misc{berkel2026tsumfree,
title = {{On $t$-sumfree sequences}},
author = {Daan van Berkel and Wieb Bosma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16843},
eprint = {2609.16843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16843},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On t-sumfree sequences
AU - Daan van Berkel
AU - Wieb Bosma
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16843
UR - https://arxiv.org/abs/2609.16843
AB - 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.
ER -