I have infinitely many sets, like:

, where is positive integer, and is a positive integer. I wrote a computer program which filtered out those elements which was not in the 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