A356851 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest positive number not occurring earlier such that a(n) shares a factor with the previous Omega(a(n)) terms.

1, 2, 4, 6, 3, 9, 12, 15, 5, 10, 20, 14, 7, 21, 28, 35, 30, 25, 40, 45, 50, 18, 22, 8, 16, 24, 26, 13, 39, 52, 65, 78, 60, 33, 11, 44, 55, 66, 70, 34, 17, 51, 68, 85, 102, 90, 38, 19, 57, 76, 95, 114, 110, 46, 23, 69, 92, 115, 138, 130, 58, 29, 87, 116, 145, 174, 150, 62, 31, 93, 124, 155, 186

Offset: 1

Author

Scott R. Shannon, Aug 31 2022

Keywords

• nonn

Comments

For n > 250 the terms are concentrated along seven lines, see the linked images. Unlike the other six lines, numbers along the second lowest line are somewhat spread out, and these terms contain all numbers with Omega(a(n)) > 1. The lowest line contains all the primes, while the upper five lines contain terms with Omega(a(n)) = 2, 3, and 4. The primes up to n=100000 occur in their natural order except for 11 and 13 which are switched. The only fixed point beyond the first two terms is 10, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(8) = 15 as Omega(15) = A001222(15) = 2, and 15 shares a factor with the previous two terms, namely a(6) = 9 and a(7) = 12.
		

Links

• Scott R. Shannon, Image of n=1..100000. The green line is a(n) = n.
• Scott R. Shannon, Image of n=1..100000 with color. Terms with Omega(a(n)) = 1, 2, 3, 4, 5, >=6 are shown in red, yellow, green, blue, violet, white respectively.

Crossrefs

Cf. A064413, A356850 (coprime to previous Omega(a(n)) terms), A001222, A093714, A336957, A000040