2,3 and 4
First we tried some examples to see if they worked, noting that 2, 3 and 4 was the first obvious answer. Then we found 14,15,16. Then we found 26,27,28. Then we looked at the sequence as it repeated and found that 12 was being added to each number.
3,4,5. The first answer was 3,4,5; then the next was 63,64,65 and we found the differences between them was 60.
We noticed that the difference was the lowest common multiple of the three numbers. (LCM of 2,3 and 4 = 12 and LCM of 3,4 and 5 is 60)
Then we tested 4,5,6 and found the LCM of these was 60, so 64,65,66 worked.
For four consecutive numbers, we started with 2,3,4,5. The LCM of these was also 60, so 62,63,64,65 worked. The pattern also worked for five consecutive numbers - 2,3,4,5,6 (LCM is 60 again, so 62,63,64,65,66 will work).