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a(n) is the smallest number whose Collatz trajectory contains n, if trajectories do not terminate at 1 but continue to cycle through 1, 4, 2, 1, 4, 2, 1, ... .
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%I #16 May 25 2024 14:42:45

%S 1,1,3,1,3,6,7,3,9,3,7,12,7,9,15,3,7,18,19,7,21,7,15,24,25,7,27,9,19,

%T 30,27,21,33,7,15,36,37,25,39,7,27,42,43,19,45,15,27,48,43,33,51,7,15,

%U 54,55,37,57,19,39,60,27,27,63,21,43,66,39,45,69,15,27

%N a(n) is the smallest number whose Collatz trajectory contains n, if trajectories do not terminate at 1 but continue to cycle through 1, 4, 2, 1, 4, 2, 1, ... .

%C a(n) = A070167(n) for n >= 5.

%C a(n) = n if 3 divides n.

%D R. K. Guy, Unsolved Problems in Number Theory, E16.

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e For n=8,

%e the trajectory of 1 is 1, 4, 2, 1, 4, ... (8 does not appear), and

%e the trajectory of 2 is 2, 1, 4, 2, 1, ... (8 does not appear), but

%e the trajectory of 3 is 3, 10, 5, 16, 8, ... (8 does appear),

%e so a(8) = 3.

%Y Cf. A070167 (sequence resulting if trajectories terminate at 1).

%K nonn

%O 1,3

%A _Ethan E. Wood_, May 13 2024

%E Edited by _Jon E. Schoenfield_, May 13 2024