What are the next several lines and how are they formed?

Do the sequences have any special feature?

Does the pattern continue indefinitely?

As I previously had suspected, this sequence has nothing to do with prime numbers.  It is a red herring, and we were all fooled by it.

The most notable feature of the sequence, besides the prime lengths, is that there is a 2 in the middle of every sequence after the first.  I further noticed that each sequence after the second contained two 3s, which seemed to split the sequences into thirds. Similarly, 4s split into fourths, 5s into fifths, etc. At first I tried thinking about factors, until I realized there was a much simpler explanation.

As pcbouhid correctly observed, each sequence is the same as the previous sequence, except with new numbers inserted.  Row N is the same as row N-1 except with a few terms with value N have been inserted.  This pattern continues.

With all this in mind, I realized that each sequence represented fractions, placed in order from least to greatest.

Consider all the proper fractions with denominators less than 6. (Well, technically, the first and last aren't proper fractions)

0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 1/4, 1/5, 1/6, 1/1
Taking only the denominators:
1, 6, 5, 4, 3, 5, 2, 5, 3, 4, 5, 6, 1

This sequence exactly matches that of row 6.

gofish seems to have gotten the answer already, since he has already posted what I assume are the correct next few sequences.  He has also provided the next few lengths of the sequences.  The first non prime is at row 10, with 33 terms.

Bob Smith presumably only took the first 6 sequences since only the first 6 follow the sequence of all primes.

Posted by Tristan on 2005-10-31 21:03:56