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 A354853 #10 Aug 03 2022 23:25:53
%S A354853 4,9,10,21,8,27,14,15,16,25,6,35,32,49,12,77,30,121,18,55,42,125,24,
%T A354853 65,64,169,20,39,70,81,28,33,128,243,22,45,256,105,26,63,512,231,34,
%U A354853 99,289,66,85,36,625,78,95,48,361,60,133,40,343,90,91,50,2197,110
%N A354853 a(1) = 4, a(2) = 9; let i = a(n-2) and j = a(n-1); a(n+1) = k 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 neither i | k nor k | i.
%C A354853 A restriction on the Yellowstone sequence A098550 analogous to A353917 regarding its relationship to A064413. This sequence exhibits phases similar to those in A353917, except between every other term instead of adjacent terms.
%C A354853 Let P = the set of distinct prime divisors of i = a(n-2), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(i, k) > 1 and let G = {P intersect Q}. Noncoprime i and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither i | k nor k | i.
%C A354853 Theorem: terms are composite. Proof: since divisibility and coprimality between i and k is prohibited and since primes must either divide or be coprime to other numbers, no primes appear in the sequence.
%C A354853 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. Hence we start the sequence with {4, 9}.
%C A354853 Theorem: squarefree semiprimes i = pq are followed by k = p^2 or k = q^2. Proof: since omega(i) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with gcd(i, k) > 1.
%C A354853 1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.
%C A354853 2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.
%C A354853 The first case implies some prime r | k yet gcd(i, r) = 1. But this would require i | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = p^e, e > 1, or k = q^m, m > 1.
%H A354853 Michael De Vlieger, <a href="/A354853/b354853.txt">Table of n, a(n) for n = 1..10000</a>
%H A354853 Michael De Vlieger, <a href="/A354853/a354853.png">Annotated log-log scatterplot of a(n)</a>, n = 1..10^4, with records in red, local minima in blue, highlighting composite prime powers in magenta, squarefree semiprimes in gold, and other squarefree numbers in green.
%t A354853 nn = 120; s = {4, 9}; state = {3, 7}; u = 4; c[_] = 0; p[_] = 2; p[2] = p[3] = 3; 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[#], c[s[[#]]]}, {s[[#]], #}] &, Length[s]]; While[Nand[c[u] == 0, CompositeQ[u]], u++]; Set[{i, j}, s[[-2 ;; -1]]]; Do[k = u; If[PrimeNu[i] == PrimeOmega[i] == 2, k = Min[Map[#^p[#] &, FactorInteger[i][[All, 1]]]], 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[PrimePowerQ@ k, p[FactorInteger[k][[1, 1]]]++]; If[k == u, While[Nand[c[u] == 0, CompositeQ[u]], u++]], {n, Length[s] + 1, nn}]; Array[a, nn]
%Y A354853 Cf. A001221, A098550, A353917.
%K A354853 nonn
%O A354853 1,1
%A A354853 _Michael De Vlieger_, Jun 23 2022