Prime order and permutation properties of eight EKG-type sequences

Abstract

We give short complete proofs of several conjectures and observations recorded in the OEIS for greedy sequences of EKG type. A381873 (EKG on the numbers with at most two distinct prime factors) is a permutation of those numbers, and its primes appear in increasing order. A251622 (share a factor with one of the two previous terms) is a permutation of the positive integers, as conjectured by Adams-Watters in 2014; the same holds for its three-term analogue A395555, and in both the primes appear in increasing order. A357963 is a permutation with its primes in natural order, which proves two of the three conjectures in the entry; the third is shown to be equivalent to the statement that the first multiple of every odd prime p is 2p. In A361133 the primes appear in natural order, and every prime p >= 5 appears in the block mp, p, rp described in the entry. A240024 (EKG on the nonprimes) is a permutation of the nonprimes. In A373998 and A373999 every prime occurs, exactly one multiple of p precedes each non-seed prime p, and the primes appear in increasing order. All proofs use one mechanism, the introduction step of Lagarias, Rains and Sloane for the EKG sequence, adapted to each rule. 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.