A382253 Triangle T(n,k) = denominator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.

1, 1, 1, 1, 1, 5, 1, 1, 7, 5, 1, 1, 3, 13, 17, 1, 1, 11, 2, 7, 13, 1, 1, 13, 19, 5, 31, 37, 1, 1, 5, 11, 29, 3, 43, 25, 1, 1, 17, 25, 11, 41, 7, 19, 65, 1, 1, 19, 7, 37, 23, 11, 4, 73, 41, 1, 1, 7, 31, 41, 17, 61, 71, 9, 91, 101, 1

Offset: 0

Author

M. F. Hasler, Apr 15 2025

Keywords

• nonn
• tabl
• frac

Comments

Since the operation n @ k := (n + k)/(1 + n*k) is commutative, it is sufficient to list only the lower half of the "multiplication table", which would otherwise be an infinite square array. This triangle lists the denominators, and A382252 lists the numerators.

Examples

			The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows:
(0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore the elements in row 0 and column 0 equal the column and row index.)
  0    1    2     3     4     5     6     7     8    Denominators of lower left
  1    1    1     1     1     1     1     1     1    triangle:  1;
  2    1   4/5   5/7   2/3   7/11  8/13  3/5  10/17             1, 1
  3    1   5/7   3/5   7/13  1/2   9/19  5/11 11/25             1, 1, 5;
  4    1   2/3   7/13  8/17  3/7   2/5  11/29  4/11             1, 1, 7, 5;
  5    1   7/11  1/2   3/7   5/13 11/31  1/3  13/41             1, 1, 3, 13, 17;
  6    1   8/13  9/19  2/5  11/31 12/37 13/43  2/7                 etc.
  7    1   3/5   5/11 11/29  1/3  13/43  7/25  5/19
  8    1  10/17 11/25  4/11 13/41  2/7   5/19 16/65
The sequence lists the denominators of the values, where denominator(x) = 1 for integers, and only for the lower left triangle of the table, by rows.
		

Crossrefs

Cf. A382252, A382257; A228564 (main diagonal), A001477 (row & col. 0 of the '@' table), A000012 (row & col. 0 & 1 of the table of denominators).

Programs

PARI

apply( {A382253(n,k=-1)= k<0&& k=n-(1+n=(sqrtint(8*n+1)-1)\2)*n/2; denominator((n+k)/(1+n*k))}, [0..66])

Formula

T(n,k) = T(k,n) for all n, k >= 0;

T(n,0) = T(0,n) = T(n,1) = T(1,n) = 1 for all n >= 0;

T(n,n) = denominator(2*n/(1+n^2)) = numerator((1+n^2)/2) = A228564(n).