[seqfan] Re: Prime Numbers Sieve and some sequences

[David Seal]

> Please see the following definition of an array A(n,k):
> 
> a(n,1)=2; every k+1 consecutive terms in each row arecoprimes.

This definition needs tightening up - my suggestion is:

Row n is the lexicographically least monotonically increasing sequence such that a(n,1)=2 and each set of n+1 consecutive terms is pairwise coprime.

In that, "lexicographically least" is needed to make it a definition rather than just a characterisation that fits many different sequences, "monotonically increasing" is needed to prevent every row being 2,1,1,1,1,..., and "pairwise coprime" is needed because otherwise it just says that the single gcd of each set of n+1 consecutive terms is 1, which would result in every row being 2,3,4,5,6,... What you're after is that any pair of different terms from the set (whether a consecutive pair or not) is coprime, and that's what "pairwise coprime" means.

I've also fixed up what appears to me to be a bit of confusion about which of n and k is the row index and which the column index.

Hope I've read the definition you intended correctly!

David


> On 26 October 2019 at 17:08 Ali Sada via SeqFan <seqfan at list.seqfan.eu> wrote:
> 
> 
> 
> 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
> 
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/