A366191 Enumeration of the rational numbers in the closed real interval [0, 1] after Cantor.

0, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 6, 2, 5, 3, 4, 1, 7, 3, 5, 1, 8, 2, 7, 4, 5, 1, 9, 3, 7, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 1, 11, 5, 7, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 1, 13, 3, 11, 5, 9, 1, 14, 2, 13, 4, 11, 7, 8, 1, 15, 3, 13, 5, 11, 7, 9

Offset: 1

Author

Peter Luschny, Oct 10 2023

Keywords

• nonn
• easy
• tabf

Comments

The rational numbers in the interval [0, 1] are listed as pairs of relatively prime integers a(2*n-1) / a(2*n).

Start with (0, 1). Then append pairs (t, n - t) where t and n - t are relatively prime positive integers and 1 <= t <= floor(n/2). Sort first by n then by t in ascending order.

Examples

			Seen as an irregular table:
   1: [0,  1],
   2: [1,  1],
   3: [1,  2],
   4: [1,  3],
   5: [1,  4], [2, 3],
   6: [1,  5],
   7: [1,  6], [2, 5], [3, 4],
   8: [1,  7], [3, 5],
   9: [1,  8], [2, 7], [4, 5],
  10: [1,  9], [3, 7],
  11: [1, 10], [2, 9], [3, 8], [4, 7], [5, 6],
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12234
• Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258, (p. 250).
• Index entries for sequences related to enumerating the rationals

Crossrefs

Cf. A352911, A333856 (numerators only).

Essentially, A182972/A182973 give the numerators/denominators separately.

Programs

Maple

A366191List := proc(upto) local C, F, n, t, count;
C := [0, 1]; count := 0:
for n from 2 while count < upto do
    F := select(t -> igcd(t, n - t) = 1, [$1..iquo(n,2)]);
    C := C, seq([t, n - t], t = F);
    count := count + nops(F) od;
ListTools:-Flatten([C]) end:
A366191List(40);

Mathematica

A366191row[n_] := If[n == 1, {0, 1}, Select[Array[{#, n - #}&, Floor[n/2]], CoprimeQ[First[#], Last[#]]&]];
Array[A366191row, 20] (* Paolo Xausa, Jan 16 2024 *)