Greedy sieve processes, and eventual prime order in the partial-mask Yellowstone family Y_S
Year: 2026
Type: article
Abstract: Many OEIS sequences are produced by one greedy rule: the next term is the least unused number that shares a factor with an earlier term (the anchor) and, possibly, is coprime to another earlier term (the mask). We treat these greedy sieve processes as one class. Without a mask we give a single theory that contains the permutation, entry-pattern and growth theorems of the EKG sequence as one instance; with a mask we classify the rules by the relation between anchor and mask (sufficient conditions only). The main new result concerns the partial-mask sequences Y_S, which interpolate between A121217 (no mask) and the Yellowstone permutation A098550 (full mask): a(n) is the least unused m with gcd(m, a(n-2)) > 1 such that no prime of a fixed set S divides both m and a(n-1). For every finite set S of primes with 2 in S we prove that all class heads grow like t + o(t), that adjacent odd terms away from prime entries number o(t), and that there is a threshold p*(S) beyond which every prime enters in the block 2p, X_p, p, Y_p, k_p p; hence only finitely many pairs of primes appear out of order. The threshold is not effective. The result does not transfer to the Yellowstone permutation itself, whose prime-order conjecture remains open. Manuscript; not submitted to a journal. The results were developed with an AI system (Anthropic Claude) and checked by independent AI referee agents; no human peer review has taken place.
Author: Haver, Tomer
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222787482
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