cp's OEIS Frontend

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

%I A366191 #23 Feb 27 2024 06:17:21
%S A366191 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,
%T A366191 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,
%U A366191 9,1,14,2,13,4,11,7,8,1,15,3,13,5,11,7,9
%N A366191 Enumeration of the rational numbers in the closed real interval [0, 1] after Cantor.
%C A366191 The rational numbers in the interval [0, 1] are listed as pairs of relatively prime integers a(2*n-1) / a(2*n).
%C A366191 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.
%H A366191 Paolo Xausa, <a href="/A366191/b366191.txt">Table of n, a(n) for n = 1..12234</a>
%H A366191 Georg Cantor, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002156806">Ein Beitrag zur Mannigfaltigkeitslehre</a>, Journal für die reine und angewandte Mathematik 84 (1878), 242-258, (p. 250).
%H A366191 <a href="/index/Ra#rational">Index entries for sequences related to enumerating the rationals</a>
%e A366191 Seen as an irregular table:
%e A366191    1: [0,  1],
%e A366191    2: [1,  1],
%e A366191    3: [1,  2],
%e A366191    4: [1,  3],
%e A366191    5: [1,  4], [2, 3],
%e A366191    6: [1,  5],
%e A366191    7: [1,  6], [2, 5], [3, 4],
%e A366191    8: [1,  7], [3, 5],
%e A366191    9: [1,  8], [2, 7], [4, 5],
%e A366191   10: [1,  9], [3, 7],
%e A366191   11: [1, 10], [2, 9], [3, 8], [4, 7], [5, 6],
%e A366191   ...
%p A366191 A366191List := proc(upto) local C, F, n, t, count;
%p A366191 C := [0, 1]; count := 0:
%p A366191 for n from 2 while count < upto do
%p A366191     F := select(t -> igcd(t, n - t) = 1, [$1..iquo(n,2)]);
%p A366191     C := C, seq([t, n - t], t = F);
%p A366191     count := count + nops(F) od;
%p A366191 ListTools:-Flatten([C]) end:
%p A366191 A366191List(40);
%t A366191 A366191row[n_] := If[n == 1, {0, 1}, Select[Array[{#, n - #}&, Floor[n/2]], CoprimeQ[First[#], Last[#]]&]];
%t A366191 Array[A366191row, 20] (* _Paolo Xausa_, Jan 16 2024 *)
%Y A366191 Cf. A352911, A333856 (numerators only).
%Y A366191 Essentially, A182972/A182973 give the numerators/denominators separately.
%K A366191 nonn,easy,tabf
%O A366191 1,6
%A A366191 _Peter Luschny_, Oct 10 2023