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).

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, 42, 25, 28, 35, 32, 40, 34, 36, 17, 38, 51, 19, 45, 57, 48, 60, 44, 46, 50, 23, 52, 69, 54, 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

Offset: 1

Author

Scott R. Shannon, Dec 12 2020

Keywords

• nonn

Comments

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.

See A339671 for a similar sequence where the prime factor rules are reversed.

Examples

			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.
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.
		

Crossrefs

Cf. A064413, A336957, A098550, A255582, A020639.

Programs

Mathematica

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 *)