A357144 Square array, A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) * f(k)) where f(m) = A002487(m)/A002487(m+1) and g is the inverse of f.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 1, 1, 4, 0, 0, 5, 32, 15, 32, 5, 0, 0, 6, 14, 6, 6, 14, 6, 0, 0, 7, 4, 7, 256, 7, 4, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 128, 63, 48, 35, 48, 63, 128, 9, 0, 0, 10, 6, 2, 1, 1, 1, 1, 2, 6, 10, 0, 0, 11, 56, 27, 2048, 47, 60, 47, 2048, 27, 56, 11, 0

Offset: 0

Author

Rémy Sigrist, Sep 15 2022

Keywords

• nonn
• tabl

Comments

The function f is a bijection from the nonnegative integers to the nonnegative rational numbers.

The positive integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the multiplicative group of positive rational numbers (f and g act as isomorphisms).

Each row (or column), except the first, is a permutation of the nonnegative integers.

Examples

			Array A(n, k) begins:
  n\k | 0   1    2   3     4     5    6    7      8    9    10   11    12
  ----+------------------------------------------------------------------
    0 | 0   0    0   0     0     0    0    0      0    0     0    0     0
    1 | 0   1    2   3     4     5    6    7      8    9    10   11    12
    2 | 0   2    8   1    32    14    4    5    128    6    56   17    16
    3 | 0   3    1  15     6     7    9   63      2   27    33   31    30
    4 | 0   4   32   6   256     2   48    1   2048   60    16   62   384
    5 | 0   5   14   7     2    35    1   47     20    3  1022  119    10
    6 | 0   6    4   9    48     1   60    3     32  510    12   13    72
    7 | 0   7    5  63     1    47    3  511     14   15    61  383    33
    8 | 0   8  128   2  2048    20   32   14  32768    4   320   26   512
    9 | 0   9    6  27    60     3  510   15      4   93    30   39   258
   10 | 0  10   56  33    16  1022   12   61    320   30   196    5  1008
   11 | 0  11   17  31    62   119   13  383     26   39     5  575     1
   12 | 0  12   16  30   384    10   72   33    512  258  1008    1   960
		

Links

• Rémy Sigrist, PARI program

Crossrefs

Cf. A002487, A054429, A354522, A355090.

Programs

PARI

See Links section.

Formula

A(n, k) = A(k, n).

A(n, 0) = 0.

A(n, 1) = n.

A(n, A054429(n)) = 1 for any n > 0.

A(m, A(n, k)) = A(A(m, n), k).

A(n, A(n-1, ... A(2, 1) ... )) = 2^(A002487(n+1)-1).