This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354720 #10 Aug 03 2022 23:25:38
%S A354720 1,2,3,10,21,4,7,6,35,8,5,12,55,9,11,15,22,25,16,45,14,27,32,33,20,81,
%T A354720 64,39,28,13,42,65,18,125,66,85,24,17,30,119,36,49,60,77,40,121,70,99,
%U A354720 50,231,128,63,26,105,169,75,52,165,182,135,56,195,154,117,44
%N A354720 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.
%C A354720 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 A354720 A restriction on the Yellowstone sequence A098550 analogous to A353916 regarding its relationship to A064413.
%H A354720 Michael De Vlieger, <a href="/A354720/b354720.txt">Table of n, a(n) for n = 1..10000</a>
%H A354720 Michael De Vlieger, <a href="/A354720/a354720.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^14, showing records in red, local minima in blue, highlighting fixed points in gold, primes in green, and composite prime powers in light green.
%t A354720 nn = 120; s = Range[3]; state = {2, 3, 4, 7}; u = 1; c[_] = 0; f[j_, k_] := Which[j == k, 5, GCD[j, k] == 1, 0, True, 1 + FromDigits[Map[Which[Mod[##] == 0, 1, PowerMod[#1, #2, #2] == 0, 2, True, 0] & @@ # &, Permutations[{k, j}]], 3]]; Array[Set[{a[s[[#]]], c[#]}, {#, s[[#]]}] &, Length[s]]; While[c[u] > 0, u++]; Set[{i, j}, s[[-2 ;; -1]]]; Do[k = u; While[Nand[c[k] == 0, MemberQ[state, f[i, k]], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[c[u] > 0, u++]], {n, Length[s] + 1, nn}]; Array[a, nn]
%Y A354720 Cf. A001221, A098550, A353916.
%K A354720 nonn
%O A354720 1,2
%A A354720 _Michael De Vlieger_, Jun 23 2022