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A339670 a(1) = 2, a(2) = 3; for n>2, a(n) = smallest number not already used that shares a prime factor with a(n-2) and has a prime factor not in a(n-1).

%I A339670 #12 Dec 12 2020 21:10:18
%S A339670 2,3,4,6,10,9,5,12,15,8,18,14,20,7,16,21,22,24,11,26,33,13,27,39,30,
%T A339670 42,25,28,35,32,40,34,36,17,38,51,19,45,57,48,60,44,46,50,23,52,69,54,
%U A339670 63,56,66,49,55,70,65,58,75,29,72,87,62,78,31,64,93,68,81,74,84,37,76,111,80,90,82
%N A339670 a(1) = 2, a(2) = 3; for n>2, a(n) = smallest number not already used that shares a prime factor with a(n-2) and has a prime factor not in a(n-1).
%C A339670 Inspired by A064413 and A336957. The terms show a similar pattern to A064413, and like that sequence they are likely a permutation of the positive integers.
%C A339670 See A339671 for a similar sequence where the prime factor rules are reversed.
%e A339670 a(4) = 4 as a(3) = 4 = 2*2 and a(2) = 2, thus a(4) must contain 2 as a prime factor but must also contain a prime factor other than 2. The lowest unused number matching these criteria is 2*3 = 6.
%e A339670 a(7) = 9 as a(6) = 10 = 2*5 and a(5) = 6 = 2*3, thus a(7) must contain 2 or 3 as a prime factor but must also contain a prime factor other than 2 and 5. The lowest unused number matching these criteria is 3*3 = 9.
%t A339670 Block[{a = {1, 2, 3}, b = {2}, c = {3}, p, k}, Do[k = 2; While[Nand[FreeQ[a, k], IntersectingQ[b, Set[p, FactorInteger[k][[All, 1]]]], Length@ Complement[p, Intersection[c, p]] > 0], k++]; AppendTo[a, k]; b = c; c = p, 73]; a] (* _Michael De Vlieger_, Dec 12 2020 *)
%Y A339670 Cf. A064413, A336957, A098550, A255582, A020639.
%K A339670 nonn
%O A339670 1,1
%A A339670 _Scott R. Shannon_, Dec 12 2020