A352911 Cantor's List: Pairs (i, j) of relatively prime positive integers sorted first by i + j then by i.

1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 4, 2, 3, 3, 2, 4, 1, 1, 5, 5, 1, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 1, 7, 3, 5, 5, 3, 7, 1, 1, 8, 2, 7, 4, 5, 5, 4, 7, 2, 8, 1, 1, 9, 3, 7, 7, 3, 9, 1, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 1, 11, 5, 7, 7, 5, 11, 1

Offset

1,4

Comments

a(2*n-1) / a(2*n) is the n-th fraction in Cantor's enumeration of the positive rational numbers. - Peter Luschny, Oct 10 2023

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..9914

Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258, (p. 250).

N. J. A. Sloane, List of the 4957 pairs (i,j) with i+j <= 127. [Note this is not a b-file.]

Index entries for sequences related to enumerating the rationals

Example

The first few pairs are, seen as an irregular triangle:
  [1, 1],
  [1, 2], [2, 1],
  [1, 3], [3, 1],
  [1, 4], [2, 3], [3, 2], [4, 1],
  [1, 5], [5, 1],
  [1, 6], [2, 5], [3, 4], [4, 3], [5, 2], [6, 1],
  [1, 7], [3, 5], [5, 3], [7, 1],
  [1, 8], [2, 7], [4, 5], [5, 4], [7, 2], [8, 1],
  [1, 9], [3, 7], [7, 3], [9, 1],
  ...

Maple

CantorsList := proc(upto) local C, F, n, t, count;
C := NULL; count := 0:
for n from 2 while count < upto do
    F := select(t -> igcd(t, n-t) = 1, [$1..n-1]);
    C := C, seq([t, n - t], t = F);
    count := count + nops(F) od:
ListTools:-Flatten([C]) end:
CantorsList(40);  # Peter Luschny, Oct 10 2023

Mathematica

A352911row[n_]:=Select[Array[{#, n-#}&, n-1], CoprimeQ[First[#], Last[#]]&];
Array[A352911row, 10, 2] (* Generates 10 rows *) (* Paolo Xausa, Oct 10 2023 *)

Prog

(Python)
from math import gcd
from itertools import chain, count, islice
def A352911_gen(): # generator of terms
    return chain.from_iterable((i, n-i) for n in count(2) for i in range(1, n) if gcd(i, n-i)==1)
A352911_list = list(islice(A352911_gen(), 30)) # Chai Wah Wu, Oct 10 2023

Crossrefs

Cf. A352909, A020652 or A038566 (i-coordinates), A020653 (j-coordinates), A366191.

Sequence in context: A286554 A352784 A037162 * A278566 A255559 A181935

Adjacent sequences: A352908 A352909 A352910 * A352912 A352913 A352914

Keyword

nonn,tabf,easy,look

Author

N. J. A. Sloane, Apr 09 2022

Status

approved