A245691 Irregular triangle of Collatz like iteration, x -> 3x, then repeat (x -> ceiling(x/2) if divisible by 3, otherwise x -> 3x) while x != 6.

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

Offset: 1

Author

K. Spage, Aug 07 2014

Keywords

• nonn
• tabf

Comments

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.

Examples

			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
		

Crossrefs

Cf. A008908, A245942.

Programs

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,", "))) }