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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with both a(n-1) and the number of divisors of a(n-1).
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%I #9 Jan 16 2023 09:10:46

%S 1,2,4,6,8,10,12,3,18,9,15,20,14,16,30,22,24,26,28,21,36,27,42,32,34,

%T 38,40,44,33,48,45,39,52,46,50,54,56,58,60,51,66,62,64,70,68,72,57,76,

%U 74,78,80,5,90,63,69,84,75,81,105,96,82,86,88,92,94,98,100,102,104,106,108,87,114,110

%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that shares a factor with both a(n-1) and the number of divisors of a(n-1).

%C The sequence is conjectured to be a permutation of the positive integers although it may take a large number of terms for the primes to appear, e.g., 17 has not occurred after 100000 terms. In the same range the only fixed points are the first two terms; it is likely no more exist although this is unknown.

%H Scott R. Shannon, <a href="/A356431/a356431.png">Image of the first 100000 terms</a>. The green line is y = n.

%e a(8) = 3 as a(7) = 12 which has six divisors, and 3 is the smallest unused number that shares a factor with both 12 and 6.

%Y Cf. A356430, A356432, A000005, A354960, A348086.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Aug 07 2022