[seqfan] Prime Numbers Sieve and some sequences

[Ali Sada]

Hi Everyone, 

 

Please see the following definition of an array A(n,k):

a(n,1)=2; every k+1 consecutive terms in each row arecoprimes.

For example, the first row, every 2 consecutive terms arecoprimes. That gives us the natural numbers (except 1.)

In the second row, each 3 consecutive terms are coprimes.

In the third row, each 4 consecutive numbers are coprimes.

And so on.

Prime numbers move to the left with each step. The seconddiagonal (and all the numbers to the left) are all primes (let’s call it thePrimes Diagonal.)  

This array gives us many sequences including:

The array itself (red by antidiagonal):

2, 3, 2, 4, 3, 2, 5, 5, 3, 2, 6, 7, 5, 3, 2, 7, 8, 7, 5, 3,2, 8, 9, 8, 7, 5, 3, 2, 9, 11, 9, 11, 7, 5, 3, 2,…..

The first composite number in each row:

4,8,8,16,16,24,24,32,32,32,45,48,48,54,64,64,64,72,80,81,90,96,105,108,108,120,128,128,128,…..

I called the numbers above "barriers"  because they separate prime numbers (on theleft) from the rest of row.

Some columns don’t have any "barriers":

4,6,8,9,11,13,14,18,20,22,24,26,27,30,33,34,36,37,……

 

Powers of 2 (4,8,16,32,…) are always "barriers."And in their last appearance they become in touch directly with seconddiagonal. Other "barriers" don’t do this. There is always a primenumber between a regular barrier (like 24,45,48,..) and the Primes Diagonal.The last appearance of the power of 2 is at (n,n+2.)

For example, 64’s last appearance is at (17,19.)

Another different thing about this sieve is that somenumbers disappear then reappear. For example, 26 disappears on the third row,then reappears on the 4th and 5th rows, then disappears forever. Maybe thesenumbers deserve a sequence on their own.

This sieve seems promising when it comes to counting thenumber of primes smaller than n, or at least the number of primes smaller thana power of 2.

 

This is a screen shot of the array.

https://justpaste.it/img/d2a04fba334538242102f31004891c38.jpg

Best,

 

Ali