Since nobody had any brilliant insights overnight, I'm going to toss some ideas out, not because I have much confidence in them, but in the hopes that they will spark another Funster. First, observe that "lex. earliest" sequences can be very hard. The lexicographically earliest sequence to satisfy some constraint might not exist for many reasons. It might require intensive backtracking. It might be very hard to prove the correctness of a given term, because subsequent backtracking could dislodge it. This, in contrast, is the easy kind of lexicographically earliest sequence. It's easy to prove that any valid start can be extended indefinitely. That means that the greedy algorithm can be used to settle the next element for all time, and no backtracking will ever be necessary. (All of this follows only after realizing that prime powers are always wrong, and must be avoided.) My other observation is that in the part of the sequence that Neil has given, there are no *tri-*composite numbers. So I will sharpen his "permutation" question and ask: does 30 ever occur? On Sat, Aug 15, 2020 at 11:56 PM Neil Sloane <njasloane@gmail.com> wrote:
Obviously this is an inverted version of the Yellowstone sequence A098550 ! The name Enots Wolley is for personal use only, it must not be mentioned in the OEIS! We frown on such made-up names.
Definition: Lexicographically earliest sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2). 1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 39, 26, 28, 63, ... The original idea was due to Scott, with a different sequence, but this is my (canonical!) version.
Could someone please prove the conjecture that this is a permutation of the set {1, all numbers with at least two distinct prime factors} ?
I can't even prove that every number 2*p (p prime) appears, or that there are infinitely many even terms (although I've found a dozen false proofs). It's a slippery problem.
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