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A352394 a(n) = n for n <= 3; let i = a(n-2) and j = a(n-1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either omega(i) or omega(k) exceed omega(m), where omega = A001221, and either i | k or k | i.

%I A352394 #13 Aug 03 2022 23:25:28
%S A352394 1,2,3,10,21,5,7,15,14,165,182,11,13,22,39,110,273,55,91,220,819,4,9,
%T A352394 20,63,260,693,26,33,130,231,65,77,195,154,3315,2926,17,19,34,57,170,
%U A352394 399,85,133,255,266,51,38,357,190,119,95,238,285,2618,3705,187,247
%N A352394 a(n) = n for n <= 3; let i = a(n-2) and j = a(n-1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either omega(i) or omega(k) exceed omega(m), where omega = A001221, and either i | k or k | i.
%C A352394 Theorem: i | k implies i < k, otherwise k | i implies i > k, a consequence of definition.
%C A352394 Theorem: Prime i implies i < k, since prime i is forced into i | k. Conversely, prime k implies i > k.
%C A352394 Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) >= p and by definition of "prime", p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited.
%C A352394 A restriction on the Yellowstone sequence A098550 analogous to A113552 regarding its relationship to A064413.
%C A352394 Conjecture: sequence is not a permutation of natural numbers. Proof sketch: Since either i | k or k | i, and defining m as the smaller of the 2 terms, as n increases, it becomes harder to reach all numbers through multiplication or division by m. Therefore it would seem that there is strong tendency for the sequence to fall into multiplicative recurrence as does A113552.
%H A352394 Michael De Vlieger, <a href="/A352394/b352394.txt">Table of n, a(n) for n = 1..5000</a>
%H A352394 Michael De Vlieger, <a href="/A352394/a352394.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^12, labeling records in red and local minima in blue, highlighting primes in green, composite prime powers in cyan, with fixed points in gold.
%t A352394 nn = 120; c[_] = False; a[1] = 1; i = a[2] = 2; j = a[3] = 3; u = 4; c[1] = c[2] = True; facs = {2}; Do[k = u; While[Nand[! c[k], Xor[And[Length@ Complement[facs, #] > 0, Divisible[i, k]], And[Length@ Complement[#, facs] > 0, Divisible[k, i]]] &[FactorInteger[k][[All, 1]]], CoprimeQ[j, k]], k++]; Set[{a[n], c[k]}, {k, True}]; i = j; j = k; facs = FactorInteger[i][[All, 1]]; If[k == u, While[c[u], u++]], {n, 4, nn}]; Array[a, nn]
%Y A352394 Cf. A001221, A098550, A113552.
%K A352394 nonn
%O A352394 1,2
%A A352394 _Michael De Vlieger_, Jun 23 2022