20 Questions Type Puzzel

[TheRealJoeFish]

This highlights an important rule when figuring out sequence puzzles: think not only about the differences between numbers, but how each difference relates to each of the numbers it's "between", and the sum of the sequence up to that point, and the like.

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Rule:

For each number in the sequence, there are two factors such that one of them is also a factor of the previous number and the other factor is 1 more or 1 less than the other factor of the previous number.

For example, in sequence 2:

1 = 1 * 1
2 = 1 * 2 (one factor is shared by the previous number, the other factor is different from the other factor of the previous number by 1)
4 = 2 * 2 (2 = 2 and the other 2 is 1 more than 1)
5 = 1 * 5 (compare to 1 * 4 = 4, the previous number)
10 = 2 * 5 (notice 5 is shared by the previous one as a factor and 2 is 1 more than 1)
12 = 2 * 6. It is also equal to 3 * 4
16 = 4 * 4 (compare to 3 * 4). It is also equal to 8 * 2
24 = 8 * 3
27 = 9 * 3

But 1 followed by 6 or 7 is not allowed. So 1 * 7 both of which share a lowest factor of 1 is are allowed.

I'm not sure I understand... 1 can't be followed by 7 because only 1*1 goes into 1 (if we're staying in integers), so the number that follows it would have to be either 1*(1-1)=0 or 1*(1+1)=2.

Or am I misunderstanding?