A341458 Unique square array T(n, k) read by antidiagonals, n, k > 0, such that A000069(T(n, k)) = A341288(A000069(n), A000069(k)).

1, 2, 2, 3, 1, 3, 4, 5, 5, 4, 5, 6, 1, 6, 5, 6, 3, 7, 7, 3, 6, 7, 4, 2, 1, 2, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 7, 4, 2, 1, 2, 4, 7, 9, 10, 17, 6, 3, 7, 7, 3, 6, 17, 10, 11, 18, 33, 5, 6, 1, 6, 5, 33, 18, 11, 12, 22, 39, 57, 4, 5, 5, 4, 57, 39, 22, 12

Offset: 1

Author

Rémy Sigrist, Feb 12 2021

Keywords

• nonn
• tabl
• base

Comments

The positive integers equipped with T form a group.

Every row (and column) is a self-inverse permutation of the positive integers.

Examples

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
  ---+--------------------------------------------------------------------------
    1|   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15   16
    2|   2   1   5   6   3   4    8    7   17   18   22   21   20   19   23   24
    3|   3   5   1   7   2   8    4    6   33   39   35   37   36   38   34   40
    4|   4   6   7   1   8   2    3    5   57   63   62   60   61   59   58   64
    5|   5   3   2   8   1   7    6    4   65   71   70   68   69   67   66   72
    6|   6   4   8   2   7   1    5    3   89   95   91   93   92   94   90   96
    7|   7   8   4   3   6   5    1    2  105  106  110  109  108  107  111  112
    8|   8   7   6   5   4   3    2    1  113  114  115  116  117  118  119  120
    9|   9  17  33  57  65  89  105  113    1   25   41   49   73   81   97  121
   10|  10  18  39  63  71  95  106  114   25    1   56   48   88   80  121   97
   11|  11  22  35  62  70  91  110  115   41   56    1   32  104  121   80   81
   12|  12  21  37  60  68  93  109  116   49   48   32    1  121  104   88   73
   13|  13  20  36  61  69  92  108  117   73   88  104  121    1   32   48   49
   14|  14  19  38  59  67  94  107  118   81   80  121  104   32    1   56   41
   15|  15  23  34  58  66  90  111  119   97  121   80   88   48   56    1   25
   16|  16  24  40  64  72  96  112  120  121   97   81   73   49   41   25    1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8385
• Rémy Sigrist, Colored representation of the table for 1 <= n, k <= 128
• Rémy Sigrist, PARI program for A341458

Crossrefs

Cf. A000069, A341288.

See A341487 and A341489 for the second and third rows, respectively.

Programs

PARI

See Links section.

Formula

T(n, k) = T(k, n).

T(n, 1) = n.

T(n, n) = 1.