20180712, 10:02  #1 
Jul 2018
1 Posts 
Sieve of what?
Has someone study previously the following structure/pattern in prime numbers?
It is similar to the Euler Sieve, but it is not, as the numbers evaluated after the square of the prime number are only those that has not been previously sieved by lower prime numbers. Any idea? a) genesis 1 (1)  > the numbers in the parenthesis indicates the pattern in the consecutive differences between the elements of the sequence, in this case the infinite set of integer numbers, from 1 to infinite. Starting in 1 it continues by adding (1) as 2 3 4 5 6 7 8 9 …. up to infinite This patter in the prime numbers will be seen up to 2² → 2 3 the total distance of the pattern is 1. b) prime number patterns b.2) using two repetitive patterns of the previous sequence, then we add to it the influence of 2. (1,1) the first two components are 2 and 3, but as 2 is the middle and it is removed then the previous expression combines to (2): 1 (2) → 3 5 7 9 11 …….. up to infinite This pattern in the prime numbers will be seen up to 3² → 3 5 7 The total pattern of the distance is 2. b.3) Now we use three repetitive patterns of the past sequence: 1 (2,2,2) As the 3 enters into action is removed then the first two patterns are combined and the final sequence remains as: 1 (4,2) → 1 5 7 11 13 17 19 21 … up to infinite, duplicating the same pattern. On the prime number list it is only visible up to 5², the square of the next prime number. The total distance of the pattern is 6 b.5) We replicate the previous pattern 5 times, having: 1 (4,2,4,2,4,2,4,2,4,2) In this case we remove 5, that is between the first 4 and 2, and 5*5, that it is placed just before the last 4. The pattern recombines to: 1 (6,4,2,4,2,4,6,2) The 5 multiples that fixed this sequence are spaced exactly as the previous sequence (4,2) 5*1 +4*5 5*5 +2*5 5*7 +4*5 5*11 ….. The output of this sequence shows the sequence of prime numbers up to 7² 1 (6,4,2,4,2,4,6,2) → 1 7 11 13 17 19 23 29 31 37 41 43 ….. up to infinitum repeating the same pattern. The total distance of the pattern is 30 b.7 ) With seven it can be done exactly as in the previous cases. 1. A replication of the previous pattern is done 7 times, 1 ( 6,4,2,4,2,4,6,2 ,6,4,2,4,2,4,6,2 ,6,4,2,4,2,4,6,2 , 6,4,2,4,2,4,6,2 , 6,4,2,4,2,4,6,2 6,4,2,4,2,4,6,2) The length of the pattern is 210 Over this pattern are introduced the modifications induced by 7, those are indicated in the previous pattern: 7*1 7*7 7*11 7*13 7*17 7*19 7* 23 7* 29 7* 31 = 217 < 210 The same sequence follows afterwards, repeating the pattern again. etc The last factor modifying is the one below the patter Once the pattern is modified taking into account the effect of 7 the sequence will provide up to 11² the list of prime numbers. The last part of the sequence will continue up to 210 but it will be modified by the 11 prime number. Once the pattern for 7 has been calculated it is possible to continue with 11. So basically each consecutive prime number modifies the previous sequence, originating a new pattern that replicates infinite times. Also this sequences fixes how the next sequence will be modified by the next prime number. Moreover it shows that the distribution of prime number follows a pattern, very simple in the very first prime numbers, and very complicated as an increasing number of prime numbers starts to modify the simpler patterns. But, well, although complicated it is a pattern 
20180712, 12:43  #2 
Aug 2006
16E2_{16} Posts 
I believe you are asking if it is known that the numbers coprime to the primorials p# are periodic with period (dividing) p#. Indeed, and this is true in general, not just for primorials. For example, you might choose to sieve out multiples of 2, 3, and 7, noting that all primes aside from 2, 3, and 7 themselves must be of one of the following 12 forms:
42n + 1 42n + 5 42n + 11 42n + 13 42n + 17 42n + 19 42n + 23 42n + 25 42n + 29 42n + 31 42n + 37 42n + 41 and so this pattern repeats over and over again, even though I've skipped over the prime 5. (It doesn't mind.) 
20190505, 21:27  #3 
Apr 2019
5×41 Posts 
I think this is essentially describing: https://en.wikipedia.org/wiki/Wheel_factorization

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