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A245691
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Irregular triangle of Collatz like iteration, x -> 3x, then repeat (x -> ceiling(x/2) if divisible by 3, otherwise x -> 3x) while x != 6.
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1
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1, 3, 2, 6, 2, 6, 3, 9, 5, 15, 8, 24, 12, 6, 4, 12, 6, 5, 15, 8, 24, 12, 6, 6, 18, 9, 5, 15, 8, 24, 12, 6, 7, 21, 11, 33, 17, 51, 26, 78, 39, 20, 60, 30, 15, 8, 24, 12, 6, 8, 24, 12, 6, 9, 27, 14, 42, 21, 11, 33, 17, 51, 26, 78, 39, 20, 60, 30, 15, 8, 24, 12
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OFFSET
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1,2
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COMMENTS
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It is conjectured that the number of steps for the trajectory to arrive at 6 is equal to the number of steps for the Collatz trajectory to arrive at 1 for the same starting value n (n>1), suggesting the length of the n-th row of the irregular array is given by A008908(n). Note that if the starting value of a trajectory in the Collatz sequence is not treated as a potential stopping value, then the conjecture would also be valid for n = 1.
Starting with x the first step in this sequence is always to multiply by 3. Thereafter if x <> 6, divide by 2 (rounding up) if x mod 3 = 0, otherwise multiply by 3. If the initial multiply-by-3 step is omitted the sequence still arrives at 6 for any starting value (conjecturally), but the length of the trajectory would no longer be the same as the length of the Collatz trajectory for starting values (n>1) that are divisible by 3.
While any odd number in the classic Collatz trajectory is immediately followed by an even number, trajectories in this sequence may contain a contiguous run of odd numbers. The trajectory starting with 27 is the lowest with more odd numbers than even numbers in its sequence.
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LINKS
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EXAMPLE
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The irregular array a(n,k) starts:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
1: 1 3 2 6
2: 2 6
3: 3 9 5 15 8 24 12 6
4: 4 12 6
5: 5 15 8 24 12 6
6: 6 18 9 5 15 8 24 12 6
7: 7 21 11 33 17 51 26 78 39 20 60 30 15 8 24 12 6
8: 8 24 12 6
9: 9 27 14 42 21 11 33 17 51 26 78 39 20 60 30 15 8 24 12 6
10: 10 30 15 8 24 12 6
11: 11 33 17 51 26 78 39 20 60 30 15 8 24 12 6
12: 12 36 18 9 5 15 8 24 12 6
13: 13 39 20 60 30 15 8 24 12 6
14: 14 42 21 11 33 17 51 26 78 39 20 60 30 15 8 24 12 6
15: 15 45 23 69 35 105 53 159 80 240 120 60 30 15 8 24 12 6
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PROG
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(PARI) { for(n=1, 15, x=n*3; print1(n, ", ", x, ", "); while(x!=6, if(x%3, x*=3, x=ceil(x/2)); print1(x, ", "))) }
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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