Two classes of modular $p$-Stanley sequences

Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successive integers that do not create a $p$-term arithmetic progression. For $p \gt 3$ prime, we give two distinct constructions for $p$-Stanley sequences which have a regular structure and satisfy certain conditions in order to be modular $p$-Stanley sequences, a set of particularly nice sequences defined by Moy and Rolnick which always have a regular structure.

Odlyzko and Stanley conjectured that the $3$-Stanley sequence generated by $\lbrace 0, n \rbrace$ only has a regular structure if $n = 3^k$ or $n = 2 \cdot 3^k$. For $p \gt 3$ we find a substantially larger class of integers $n$ such that the $p$-Stanley sequence generated from $\lbrace 0, n \rbrace$ is a modular $p$-Stanley sequence and numerical evidence given by Moy and Rolnick suggests that these are the only n for which the $p$-Stanley sequence generated by $\lbrace 0, n \rbrace$ is a modular $p$-Stanley sequence. Our second class is a generalization of a construction of Rolnick for $p = 3$ and is thematically similar to the analogous construction by Rolnick.

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Received 21 July 2017

Published 27 September 2019