A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d*(m-1),m*(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.

1, 1, 2, 1, 0, 3, 1, 0, 0, 4, 1, 0, 2, 0, 5, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, 3, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 14

Offset: 0

Author

Omar E. Pol, Jun 16 2025

Keywords

• nonn
• tabl

Comments

Given a number d whose position is (n,k) so the next number d is in the position (n+d,k+d-1). In other words: we can find the next number d by walking d steps south and then walking d-1 steps east. Hence the Manhattan distance between two nearest numbers d is 2*d - 1.

By definition the m-th number d is in the position (d*(m-1),m*(d-1)) thus all the numbers d are on the same straight line.

The positive terms in the row n >= 1 are the divisors of n in increasing order. Hence the number of positive terms in row n is A000005(n).

The positive terms in the column k >= 1 are 1 plus the divisors of k in decreasing order. Hence the number of positive terms in the column k is A000005(k).

In the row n >= 1, two conjugate divisors of n are equidistant from the position (n,n-1). That position is in the same column that contains to the number n in the row 0.

In the column k >= 1, two conjugate (1 plus divisor)'s of k are equidistant from the position (k,k+1).

On the other hand we can find the divisors of n on a curve which starts at A(n-1,0) = 1 and ends at A(0,n-1) = n. Hence the number of positive terms in the curve (n-1) is A000005(n). Here that curve is called "curve of divisors of n".

Note that every divisor of n less to n on the curve is also a divisor of a number less to n in a row. Hence every divisor of n in a row is also a divisor on the curve of divisors of a number greater than n.

The position of two conjugate divisors of n on the curve is orthogonal and equidistant to the main diagonal of the array.

Curves never touch each other. See the curve and the row both with the divisors of 12 in the Example section.

Drawing all the curves the 2D structure is compatible with a 3D model where in orthogonal position is the arc diagram of A000005. See Links section.

The main diagonal is an irregular triangle read by rows in which row r lists r together with 2*r - 1 zeros, r >= 1.

Examples

			The corner 9 X 9 of the square array is as shown below:
.
   \k   0 1 2 3 4 5 6 7 8
   n\ _ _ _ _ _ _ _ _ _ _
     |
   0 |  1 2 3 4 5 6 7 8 9
   1 |  1 0 0 0 0 0 0 0 0
   2 |  1 0 2 0 0 0 0 0 0
   3 |  1 0 0 0 3 0 0 0 0
   4 |  1 0 0 2 0 0 4 0 0
   5 |  1 0 0 0 0 0 0 0 5
   6 |  1 0 0 0 2 0 3 0 0
   7 |  1 0 0 0 0 0 0 0 0
   8 |  1 0 0 0 0 2 0 0 0
  ...
.
The corner 25 X 25 of the square array without the zeros is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   0 |  1 2 3 4 5 6 7 8 9 ...
   1 |  1
   2 |  1   2
   3 |  1       3
   4 |  1     2     4
   5 |  1               5
   6 |  1       2   3       6
   7 |  1                       7
   8 |  1         2       4         8
   9 |  1               3               9
  10 |  1           2           5           10
  11 |  1                                       11
  12 |  1             2     3   4     6             12
  13 |  1                                               13
  14 |  1               2                   7
  15 |  1                       3       5
  16 |  1                 2           4           8
  17 |  1
  18 |  1                   2       3           6       9
  19 |  1
  20 |  1                     2             4   5
  21 |  1                               3               7
  22 |  1                       2
  23 |  1
  24 |  1                         2         3     4
  ...
.
The corner 25 X 25 of the square array without the zeros with the row and the curve of the divisors of 12 is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
     |
   0 |                        12
   1 |
   2 |
   3 |
   4 |
   5 |
   6 |                      6
   7 |
   8 |                    4
   9 |                  3
  10 |              2
  11 |  1
  12 |  1             2     3   4     6             12
  ...
.
The position of the conjugate divisors of 12 on the curve is symmetric respect to the main diagonal of the square array.
The position of the conjugate divisors of 12 in row 12 is symmetric respect the position (12,11). That position is in the same column that contains to the number 12 in the row 0.
		

Links

• Omar E. Pol, Illustration of initial terms of A000005 (arc diagram)

Crossrefs

Row sums give A000203, n >= 1.

Column sums give A007503 = A000203 + A000005, k >= 1.

Row 0 gives A000027.

Column 0 gives A000012.

Cf. A027750, A056538, A208460.