A340779 a(1)=1, a(2)=2; for n>=3, a(n) = the closest number to a(n-1) that has not occurred earlier and has at least one common factor with a(n-1), but none with a(n-2). In case of a tie, choose the smaller.

1, 2, 6, 15, 35, 28, 26, 39, 33, 22, 20, 45, 51, 34, 38, 57, 63, 56, 58, 87, 93, 62, 68, 85, 75, 72, 74, 185, 175, 168, 166, 415, 405, 402, 404, 505, 495, 492, 494, 481, 407, 396, 394, 985, 975, 972, 974, 2435, 2425, 2328, 2326, 5815, 5805, 5802, 5804, 7255, 7245, 7242, 7244, 9055

Offset: 1

Author

Scott R. Shannon, Jan 21 2021

Keywords

• nonn

Comments

The sequence uses a similar selection rule to the Enots Wolley sequence A336957 but instead of choosing the smallest number that has not occurred earlier that has a common factor with a(n-1) and no common factor with a(n-2), the number closest to a(n-1) that satisfies these rules is selected for a(n). If two such numbers are the same distance from a(n-1) then the smaller is chosen. Like A336957 for the sequence to continue a(n) must always have a prime factor not in a(n-1), thus a(n) cannot be a prime or a prime power.

The sequence grows sporadically with n, showing regions of little growth followed by a large jump due to the next term being the multiple of a large prime of the previous term. However due to the overall rapid increase in the terms it is very unlikely any fixed points exist.

Examples

			a(5) = 35 as a(4) = 15 = 3*5 and a(3) = 6 = 2*3, thus a(5) must be a multiple of 5 while not being a multiple of 3, and must have a prime factor other than 5. The smallest unused number closest to 15 satisfying these criteria is 35.
a(6) = 28 as a(5) = 35 = 5*7 and a(4) = 15 = 3*5, this a(6) must be a multiple of 7 while not being a multiple of 5, and must have a prime factor other than 7. The smallest number satisfying these criteria is 14. However 28 also does and is only 7 away from a(5), while 14 is 21 away, thus 28 is chosen. This is the first term that differs from A336957.
		

Crossrefs

Cf. A336957, A098550, A340783, A340807, A064413, A121216.