A337181 a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest number not already used that is a multiple of at least one prime factor of both a(n-1) and a(n-2).

1, 2, 4, 8, 12, 6, 9, 18, 24, 16, 20, 10, 25, 30, 15, 27, 36, 42, 14, 21, 28, 48, 32, 40, 44, 22, 52, 26, 56, 60, 35, 45, 50, 54, 64, 66, 68, 34, 72, 51, 63, 81, 84, 78, 39, 90, 65, 70, 75, 80, 96, 76, 38, 88, 92, 46, 100, 104, 108, 102, 99, 33, 117, 126, 91, 49, 98, 112, 116, 58, 120

Offset: 1

Author

Scott R. Shannon, Jan 28 2021

Keywords

• nonn

Comments

As each term must have at least two prime factors no term, other than the initial 2, can be prime.

Examples

			a(4) = 8 as the factors of a(4-2) = a(2) = 2 and a(4-1) = a(3) = 4 = 2*2, thus a(4) must be the minimum unused multiple of 2*2 = 4, which is 8.
a(6) = 6 as the factors of a(6-2) = a(4) = 8 = 2*2*2 and a(6-1) = a(5) = 12 = 2*2*3, thus a(4) must be the minimum unused multiple of 2*2 = 4 or 2*3 = 6. As 4 has been used a(6) = 6.
a(13) = 25 as the factors of a(13-2) = a(11) = 20 = 2*2*5 and a(13-1) = a(12) = 10 = 2*5, thus a(13) must be the minimum unused multiple of 2*2 = 4, 2*5 = 10, or 5*5 = 25. As 4,8,10,12,16,20,24 have been used, a(13) = 25.
		

Links

• Scott R. Shannon, Image of the terms for n=1..500000. The green line is a(n) = n, indicating the other lines are slightly curved downward.

Crossrefs

Cf. A064413, A336957, A098550, A270139.