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 > 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 {0, n} only has a regular structure if n = 3^kor n = 2·3^k. For p > 3 we find a substantially larger class of integers n such that the p-Stanley sequence generated from {0, n} 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 {0, n} 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.