I found a sequence that can be iterated to create a list of primes. It can certainly be considered a sieve, but the interesting part of the sequence is the underlying symmetry that seems to guide the distribution of primes within the natural numbers. The sequence contains 1 and misses 2,3,and 5. If any one is interested, the reason 2,3,5 are missed is fascinating to me and I would be happy to elaborate (it is related to phi (0.5+sqrt(5)/2). Here is the sequence:

0.5, 1.5, 1.0, 0.5, 1.0, 0.5, 1.0, 1.5

If you start with a base of -0.5 then sum the sequence you have:

0, 1.5, 2.5, 3.0, 4.0, 4.5, 5.5, 7.0, Then iterate 7.5, 9.0, 10.0, 10.5, 11.5, 12.0, 13.0, 14.5 …

Go back and multiply by 4 and add 1 to each term:

1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59 ….

Note the red 49 (which is obviously not prime). Yet, if you continue through the sequence, you will find that the result is primes and multiples of primes greater than 7. Interestingly, the sequence can be advanced to then give all prime multiples of 7, then prime multiples of 11 and so on to infinity (I believe). By "advance the sequence" I am referring to the following matrix:

1) 0.5, 1.5, 1.0, 0.5, 1.0, 0.5, 1.0, 1.5

2) 1.5, 1.0, 0.5, 1.0, 0.5, 1.0, 1.5, 0.5

3) 1.0, 0.5, 1.0, 0.5, 1.0, 1.5, 0.5, 1.5

4) 0.5, 1.0, 0.5, 1.0, 1.5, 0.5, 1.5, 1.0

5) 1.0, 0.5, 1.0, 1.5, 0.5, 1.5, 1.0, 0.5

6) 0.5, 1.0, 1.5, 0.5, 1.5, 1.0, 0.5, 1.0

7) 1.0, 1.5, 0.5, 1.5, 1.0, 0.5, 1.0, 0.5

8) 1.5, 0.5, 1.5, 1.0, 0.5, 1.0, 0.5, 1.0

Sequence (1) gives all primes and multiples of primes greater than 7.

Subtract sequence (2) from sequence (1) to eliminate all primes multiples of 7.

Subtract sequence (3) from sequence (1) to eliminate all primes multiples of 11; and so on. A simple calculation places the start of each new sequence in the correct place for subtraction to work. I think of it as a phase shift. A wave interpretation is possible which would include a base, phase shift, and change in frequency for movement through waves 1-8. I actually used this interpretation to check the process for primes less than 1 million.

At the end of sequence 8, add 30 to sequence 1 and continue on to infinity adding multiples of 30 with each iteration of the 8 sequences.

It is easier to get an idea of how the sequence works by viewing a website I created to get a visual feel for the process. It is www.GPrimes.com. I suggest you simply click each prime sequence in the header and watch the sequence labeled in red move through the process.

It may seem that the sequence is arbitrary. It is not. There is a geometric interpretation that I am happy to share if anyone is interested. Again, the most interesting thing about the sequence to me, is not the ability to print primes, but rather, the symmetry of the system that in my opinion guides the distribution of primes within the natural numbers.

I like to include a small disclaimer with my posts in a legitimate science forum like this. I am not a mathematician. Rather, I am a medical doctor that likes to play around with number theory and theoretical physics. If this is old information, please forgive me. On the other hand, I think the profession benefits from people like me that, being an outsider, tend to think out of the box a little more. That can be useful.