I have infinitely many sets, like:

$\displaystyle $a_l:= ]2 \cdot k \cdot 3^l + 3^l; 2 \cdot k \cdot 3^l + 2 \cdot 3^l [$$, where $\displaystyle $l$$ is positive integer, and $\displaystyle $k$$ is a positive integer. I wrote a computer program which filtered out those elements which was not in the $\displaystyle $a_l$$ sets.

The result is:

{6,7,8,9

18,19,20,21

24,25,26,27

54,55,56,57

60,61,62,63

72,73,74,75

78,79,80,81}

.... and so on. I observed this sequence has a period (12,6,30,6). But I have no idea how to prove it. Can you help me? Thanks.

update: k is also a positive integer