Should this array be submitted as new sequence or just comments on A020882?

[Ali Sada]

Hi everyone,

Hope all is well. Please see the array below. The numbers and their frequencies are exactly the same as A020882. I would really appreciate it if you could tell me whether a new sequence is warranted or if the comments are suitable for A020882.
A triangular array A(n,k) read by rows such that  A(n,k)= 1+4(A000217(n)+ A000217(k-1))

5;
13, 17; 
25, 29, 37;
41, 45, 53, 65; 
61, 65, 73,  85, 101; 
85, 89, 97, 109, 125, 145;

Comments:
A prime appears only once in this sequence.
A number appears more than once if and only if it’s a product of two or more distinct primes that appear in this sequence.
If R is a product of k distinct primes that appear in this sequence, then R appears at least k times.

Best,

Ali

[Ali Sada]

Correction: Please ignore the "and only if" in the second comment. 

[M. F. Hasler]

On 5 Sept. 2026   22:46, Ali Sada wrote:

Please see the array below. The numbers and their frequencies are exactly the same as A020882.

I don't think so, e.g., the number 45 never appears in  A020882, but in yours:

5;
13, 17; 
25, 29, 37;
41, 45, 53, 65; 
61, 65, 73,  85, 101; 
85, 89, 97, 109, 125, 145;

Comments:
A prime appears only once in this sequence.
A number appears more than once if and only if it’s a product of two or more distinct primes that appear in this sequence.
If R is a product of k distinct primes that appear in this sequence, then R appears at least k times.

In general, if a different formula/procedure (really) gives the same sequence, then it should be added as comment rather than a new sequence. 

There are many (even important) sequences where the terms have different interprestions and/or can be produced with completely different methods, in the same or possibly in another order.
This is usually stated in comments: "Also, ...", giving the other method/interpretation and precisions ("in [what] order", "with[out] repetitions" , ...)

But even if its not exactly the same, you might add a comment (e.g. here in  A020882)
"very similar to ...., they first differ..." or "except for 45, ..., which appear ....[where?] but not in this sequence."

- Maximilian

[Ali Sada]

Thank you very much, Maximilian. I really appreciate your help here, as always.

 

This sequence is beautiful. I spent the long weekend studying it and it was worth every second! It contains all 4k+1 primes and connects the positions and repetitions of the sum of two square numbers directly to their prime factorization. Repeated appearances reveal that a number is composite and can even be used to recover its factors. I’m trying  now to find a “counterpart” for 4k+3 primes.

Best,

Ali

P.S. My first attempt to respond to this email has failed and I don't know why. 

On Tue, Sep 8, 2026 at 1:00 PM M. F. Hasler <oe...@dsi972.fr> wrote:

PS: to be precise, the triangle doesn't exist, but it's image/range is A339952.

Le mar. 8 sept. 2026 à 12:36, M. F. Hasler <oe...@dsi972.fr> a écrit :

I tentatively added, in https://oeis.org/draft/A020882:

Ali Sada observed that the terms are to a large extend the same as in the triangle T(n,k) = 1 + 4(A000217(n)+A000217(k)), n,k > 0, i.e., 1 + 4*(sums of two positive triangular numbers), but e.g., T(4,2) = 45, T(7,1) = 117, T(7,5) = 153, ... never appear in this sequence. - M. F. Hasler, Sep 08 2026

Please feel free to change or add an XREF if you plan to submit that triangle separately.

(I checked that it does not yet exist.)

- Maximilian