Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Dec 12 2020 21:10:18
%S 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 42,25,28,35,32,40,34,36,17,38,51,19,45,57,48,60,44,46,50,23,52,69,54,
%U 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 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 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 See A339671 for a similar sequence where the prime factor rules are reversed.
%e 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 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 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 Cf. A064413, A336957, A098550, A255582, A020639.
%K nonn
%O 1,1
%A _Scott R. Shannon_, Dec 12 2020