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 A352768 #20 Apr 02 2022 12:00:00
%S A352768 1,2,6,4,8,10,12,14,16,18,20,22,15,74,178,24,26,28,21,7,30,111,3,9,27,
%T A352768 33,36,39,40,158,32,34,38,42,35,44,316,46,48,50,52,45,194,478,54,56,
%U A352768 58,60,62,64,66,55,11,70,63,49,72,77,298,78,68,76,51,254,75,84,57,69,81,87,90,93,96,98
%N A352768 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with both a(n-2)+a(n-1) and a(n-2)*a(n-1).
%C A352768 In the first 250000 terms the lowest number not to appear is 811; it is likely this sequence is a permutation of the positive integers. In this range the fixed points are 1, 2, 4, 35, 6731, 167821. It is possible more exist although this is unknown. In the same range the longest runs of consecutive even terms and odd terms is 29 and 33 respectively. This suggests such runs are likely arbitrarily long.
%H A352768 Michael De Vlieger, <a href="/A352768/b352768.txt">Table of n, a(n) for n = 1..10000</a>
%H A352768 Michael De Vlieger, <a href="/A352768/a352768_1.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^16, showing records in red, local minima in blue, fixed points in gold, and highlighting primes in green.
%H A352768 Scott R. Shannon, <a href="/A352768/a352768.png">Image of the first 250000 terms</a>. The green line is y = n.
%e A352768 a(3) = 6 as a(1)+a(2)=3, a(1)*a(2)=2, and 6 is the smallest unused number that shares a factor with both 3 and 2.
%t A352768 nn = 74; c[_] = 0; a[1] = c[1] = 1; a[2] = c[1] = 2; u = 1; While[c[u] > 0, u++]; Do[k = u; While[Nand[c[k] == 0, GCD[#1, k] > 1, GCD[#2, k] > 1], k++] & @@ {#1 + #2, #1 #2} & @@ {a[i - 2], a[i - 1]}; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 1, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger_, Apr 02 2022 *)
%Y A352768 Cf. A351001, A349493, A251622.
%K A352768 nonn
%O A352768 1,2
%A A352768 _Scott R. Shannon_, Apr 02 2022