OFFSET
1,2
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.
EXAMPLE
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
PROG
(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, ", "))) }
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
K. Spage, Aug 07 2014
STATUS
approved