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

1,6

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.

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

Example

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],
  ...

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 *)

Crossrefs

Cf. A352911, A333856 (numerators only).

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

Sequence in context: A292587 A336571 A334032 * A097283 A334033 A339564

Adjacent sequences: A366188 A366189 A366190 * A366192 A366193 A366194

Keyword

nonn,easy,tabf

Author

Peter Luschny, Oct 10 2023

Status

approved