The Permutation Property of Greedy Sequences under a Density Condition on the Forbidden Set: Applications to OEIS A121878 and A329405

Let B be a set of positive integers, called the forbidden set. Consider the greedy sequence obtained by repeatedly choosing the smallest unused positive integer whose sum with each of the previous at most r terms does not belong to B. If B has natural density β, then under the condition rβ < 1/2, we show that this sequence is a permutation of the positive integers. We apply this general theorem to OEIS A121878 and A329405, proving in each case the previously conjectured permutation property. This is an English translation of the Japanese original published on Zenodo: DOI 10.5281/zenodo.22957252.