slug: oeis-a363956-positive-permutation bibkey: shannon2023a363956 doi: null url: https://oeis.org/A363956 triage: theorem motivation_gids:
- D5/S3/Arith/OmegaGreedyPermutation
The permutation conjecture for A363956
Problem
The sequence is conjectured to be a permutation of the positive integers.
The sequence begins with 1 and 2. Thereafter take the smallest positive unused multiple of the omega-th prime, where omega counts distinct prime factors of the previous term.
Motivation
This Tier 1 OEIS conjecture is asymptotic. Missing primes in a finite prefix do not decide it. The task is surjectivity of the actual greedy recursion.
Gap
A363956 and the related A363504/A351495 entries state their permutation claims as conjectures without proofs. No priority is claimed for the repository proof.
Route
Infinitely activated queues exhaust all positive multiples. Finiteness of the queue-2 activations would make the entire prime-output set finite, hence all selected queues finite in number. One infinite queue then forces unbounded distinct-factor counts, contradiction. The resulting coverage of even numbers forces every prime queue by varying the exponent of 2 at fixed prime support.
Falsifier
A positive integer proved to be absent would refute surjectivity. A proof that some prime queue is activated only finitely often would contradict the stronger intermediate result. Finite prefix omissions provide neither witness.
Evidence
D5/S3/Arith/OmegaGreedyPermutation.lean proves sequence_initial, sequence_positive, sequence_injective, sequence_greedy and a363956_surjective. The definition uses a finite used-set recursion and a natural infimum. Fairness and surjectivity are derived, never assumed.
Triage
theorem. The full positive-integer permutation question is answered by unbounded proofs of surjectivity and injectivity.
ASSUMED-UNVERIFIED
Linked data, plots and revision histories were not opened. No assertion is made about rates, fixed points or the multiplicity-count variant A363504. The match between the Lean rule and OEIS prose is a source-fidelity assessment.