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

0,8

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.

Links

Table of n, a(n) for n=0..90.

Rémy Sigrist, PARI program

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).

Example

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

Prog

(PARI) See Links section.

Crossrefs

Cf. A002487, A054429, A354522, A355090.

Sequence in context: A341317 A101164 A229079 * A351786 A329331 A355664

Adjacent sequences: A357141 A357142 A357143 * A357145 A357146 A357147

Keyword

nonn,tabl

Author

Rémy Sigrist, Sep 15 2022

Status

approved