[seqfan] Should this array be added to the OEIS?

Thu Feb 6 03:50:08 CET 2020

```Hi Everyone,

I was working on an algorithm but I found out that there is already a good amount of literature about it in the OEIS.

The algorithm takes a pair (n, n+1), divides the even element by 2, and adds the result to the odd element, then repeats until we go back to the original pair.

If the order of the pair’s elements was not important, the number of steps to go back to the original pair would be:
1, 3, 5, 4, 9, 11, 9, 5, 12, 12, 7, 23, 8, 20, 29, 6, 33, 35, 20, 39, 41, 28, 12, 36, 15, 51  (A294673).

If the order was important, the number of steps would be:
2, 3, 10, 4, 18, 11, 18, 5, 12, 12, 14, 23, 8, 20, 58, 6, 66, 35, 20, 39, 82, 28, 12, 36, 30, 51 (A274299).
With A274299(n) equals either A294673(n) or 2*A294673(n).

I created arrays for both versions, with odd numbers as columns, and even numbers as rows. A(i,j) is the number of steps the algorithms takes.
https://justpaste.it/785m6
https://justpaste.it/4m2ff

The diagonals of these two arrays are A294673 and A274299.
The first row in the first array is A003558 (or A216066).
The first row of the second array is A002326.

The patterns in the arrays seem nice. If j+2 is a prime, the antidiagonal from A(1,j) to A(j+1,1) contains the same number of steps.
If j+2 is composite, we find discrepancies. The discrepancies happen when i and j share a factor.

For example, the antidiagonal from (1,15) to (16,1) is {8, 8, 8, 8, 8, 8, 8, 8}.
While the antidiagonal from (1,25) to (26,1) is {18, 18, 6, 18, 18, 6, 18, 18, 2, 18, 18, 6, 18}.

I was wondering if one of these two arrays should be added to the OEIS.

Best,

Ali

```