A357144 - OEIS

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

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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 A370049

Adjacent sequences: A357141 A357142 A357143 * A357145 A357146 A357147

KEYWORD

nonn,tabl

AUTHOR

Rémy Sigrist, Sep 15 2022

STATUS

approved