%I #19 Sep 05 2021 11:27:12

%S 1,2,6,15,35,28,26,39,33,22,20,45,51,34,38,57,63,56,58,87,93,62,68,85,

%T 75,72,74,185,175,168,166,415,405,402,404,505,495,492,494,481,407,396,

%U 394,985,975,972,974,2435,2425,2328,2326,5815,5805,5802,5804,7255,7245,7242,7244,9055

%N 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.

%C 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.

%C 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.

%e 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.

%e 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.

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

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jan 21 2021