A333856 -id:A333856 - cp's OEIS frontend

A182972 Numerators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

1, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 2, 4, 1, 3, 1, 2, 3, 4, 5, 1, 5, 1, 2, 3, 4, 5, 6, 1, 3, 5, 1, 2, 4, 7, 1, 3, 5, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 5, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 3, 7, 9, 1, 2, 4, 5, 8, 10, 1, 3, 5, 7, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 5, 7, 11, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 1, 3, 5, 7, 9, 11

Offset: 1

William Rex Marshall, Dec 16 2010

A023022(n) and A245677(n) give number and numerator of sum of fractions a(k)/A182973(k) such that a(k) + A182973(k) = n. - Reinhard Zumkeller, Jul 30 2014

			Positive fractions < 1 listed by increasing sum of numerator and denominator, and by increasing numerator for equal sums:
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
1/16 2/15 3/14 4/13 5/12 6/11 7/10 8/9
1/17 5/13 7/11
1/18 2/17 3/16 4/15 5/14 6/13 7/12 8/11 9/10
1/19 3/17 7/13 9/11
(this is A182972/A182973).

S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
R. K. Guy, Unsolved Problems in Number Theory (UPINT), Section D11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.
Index entries for sequences related to enumerating the rationals

Cf. A182973 (denominators), A366191 (interleaved).
Cf. A020652, A038567, A182974, A182975, A182976.
Cf. A023022, A245677, A245678, A245718.
Essentially the same as A333856.

a182972 n = a182972_list !! (n-1)
a182972_list = map fst $ concatMap q [3..] where
   q x = [(num, den) | num <- [1 .. div x 2],
                       let den = x - num, gcd num den == 1]
-- Reinhard Zumkeller, Jul 29 2014

t1:=[];
for n from 2 to 40 do
t1:=[op(t1),1/(n-1)];
for i from 2 to floor((n-1)/2) do
   if gcd(i,n-i)=1 then t1:=[op(t1),i/(n-i)]; fi; od:
od:
t1;

t1={}; For[n=2, n <= 40, n++, AppendTo[t1, 1/(n-1)]; For[i=2, i <= Floor[(n-1)/2], i++, If[GCD[i, n-i] == 1, AppendTo[t1, i/(n-i)]]]]; t1 // Numerator // Rest (* Jean-François Alcover, Jan 20 2015, translated from Maple *)

program a182972;
var
  num,den,n: longint;
function gcd(i,j: longint):longint;
begin
  repeat
    if i>j then i:=i mod j else j:=j mod i;
  until (i=0) or (j=0);
  if i=0 then gcd:=j else gcd:=i;
end;
begin
  num:=1; den:=1; n:=0;
  repeat
    repeat
      inc(num); dec(den);
      if num>=den then
      begin
        inc(den,num); num:=1;
      end;
    until gcd(num,den)=1;
    inc(n); writeln(n,' ',num);
  until n=100000;
end.

from itertools import count, islice
from math import gcd
def A182972_gen(): # generator of terms
    return (i for n in count(2) for i in range(1,1+(n-1>>1)) if gcd(i,n-i)==1)
A182972_list = list(islice(A182972_gen(),10)) # Chai Wah Wu, Aug 28 2023

Corrected by William Rex Marshall, Aug 12 2013

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

Original entry on oeis.org

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

Peter Luschny, Oct 10 2023

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.

			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],
  ...

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

Cf. A352911, A333856 (numerators only).

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

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);

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

A333857 Positive odd numbers b with an unequal number of odd and even elements of the restricted residue system of the mod* congruence of Brändli and Beyne (numbers b ordered increasingly).

Original entry on oeis.org

1, 21, 33, 57, 63, 69, 77, 93, 99, 129, 133, 141, 147, 161, 171, 177, 189, 201, 207, 209, 213, 217, 231, 237, 249, 253, 279, 297, 301, 309, 321, 329, 341, 363, 381, 387, 393, 399, 413, 417, 423, 437, 441, 453, 469, 473, 483, 489, 497, 501, 513, 517, 531, 537, 539, 553, 567, 573, 581, 589, 597

Offset: 1

Wolfdieter Lang, Jun 26 2020

nonn

For the modified congruence modulo n of Brändli and Beyne, called mod* n, see the link. See also the comments in A333856 for this reduced residue system mod* n, called RRS*(n), for n >= 1.
The odd members of RRS*(n) are denoted by RRS*odd(n), similarly, RRS*even(n) are the even elements of RRS*(n). E.g., RRS*odd(5) = {1} and RRS*even(5) = {2}. Therefore the odd number 5 can be called balanced in the reduced mod* system, because #RRS*odd(5) = 1 = #RRS*even(5).
All even numbers are unbalanced because RRS*(2*m) has only odd members, for m >= 1.
b = 1, with RRS*(1) = {0} is unbalanced, and for odd numbers b >= 3 to be balanced one needs integer phi(b)/4 because #RRS*(b) = phi(b)/2 (phi = A000010). The odd integers >= 3 with integer phi(b)/4 are given in A327922. The present sequence gives, for n >= 2, a proper subset of A327922 consisting of odd numbers b with an unequal number of odd and even elements (unbalanced odd b). Therefore, the condition phi(b)/4 integer for odd b is necessary but not sufficient for such odd b in the reduced mod* system.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.

Cf. A038566 (RRS(n)), A333856 (RRS*(n)).

Cf. A000010, A327922.

RRS(n) = select(x->gcd(n, x)==1, [1..n]); \\ A038566
RRSstar(n) = if (n<=2, [n-1], my(r=RRS(n)); Vec(r, #r/2)); \\ A333856
isok(k) = if ((k%2) && ((k==1) || denominator(eulerphi(k)/4)==1), my(v=RRSstar(k)); #select(x->((x%2) == 1), v) != #select(x->((x%2) == 0), v)); \\ Michel Marcus, Sep 17 2023

This sequence gives the increasingly ordered positive odd integers b from A327922 such that the reduced residue system RRS*(b) does not have the same number of odd and even elements, for n >= 1, The odd number b is then called unbalanced.

More terms from Michel Marcus, Sep 17 2023

A334428 Irregular triangle read by rows: row n gives the members of the smallest nonnegative reduced residue system in the modified congruence modulo 2n - 1 by Brändli and Beyne, called mod(2*n - 1).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jun 27 2020

The length of row n is A072451(n) = A055034(2*n-1), for n >= 1.
See the Brändli-Beyne link, and A333856 for the definition and some examples of this mod* system.
This reduced residue system mod* (2*n - 1) will be called RRS*(2*n - 1).
Compare this table with the one for the reduced residue system modulo 2*n - 1 (called RRS(2*n - 1) = A038566(2*n - 1), but with A038566(1) = 0). For n >= 2 RRS*(2*n-1) consists of the first half of the entries of RRS(2*n - 1).
The modular arithmetic is multiplicative but not additive for mod*. See A333856 for examples.

			The irregular triangle T(n, k) begins (b = 2*n - 1):
n    b \k  1 2 3 4 5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
---------------------------------------------------------------
1    1:    0
2    3:    1
3    5:    1 2
4    7:    1 2 3
5    9:    1 2 4
6   11:    1 2 3 4 5
7   13:    1 2 3 4 5  6
8   15:    1 2 4 7
9   17:    1 2 3 4 5  6  7  8
10  19:    1 2 3 4 5  6  7  8  9
11  21:    1 2 4 5 8 10
12  23:    1 2 3 4 5  6  7  8  9 10 11
13  25:    1 2 3 4 6  7  8  9 11 12
14  27:    1 2 4 5 7  8 10 11 13
15  29:    1 2 3 4 5  6  7  8  9 10 11 12 13 14
16  31:    1 2 3 4 5  6  7  8  9 10 11 12 13 14 15
17  33:    1 2 4 5 7  8 10 13 14 16
18  35:    1 2 3 4 6  8  9 11 12 13 16 17
19  37:    1 2 3 4 5  6  7  8  9 10 11 12 13 14 15 16 17 18
20  39:    1 2 4 5 7  8 10 11 14 16 17 19
...
-----------------------------------------------------------
For n = 5 (b = 9) see the example in A333856.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.

Cf. A055034, A072451, A038566, A333856.

Array[Function[{m, b}, Select[Range[1, m], GCD[#, b] == 1 &] /. {} -> {0}] @@ {# - 1, 2 # - 1} &, 16] // Flatten (* Michael De Vlieger, Jun 27 2020 *)

T(1, 1) = 0, T(n, k) = A038566(2*n - 1, k) for k = 1, 2, ..., A072451(n), for n >= 2.

cp's OEIS Frontend

A182972 Numerators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A333857 Positive odd numbers b with an unequal number of odd and even elements of the restricted residue system of the mod* congruence of Brändli and Beyne (numbers b ordered increasingly).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A334428 Irregular triangle read by rows: row n gives the members of the smallest nonnegative reduced residue system in the modified congruence modulo 2*n - 1 by Brändli and Beyne, called mod*(2*n - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A334428 Irregular triangle read by rows: row n gives the members of the smallest nonnegative reduced residue system in the modified congruence modulo 2n - 1 by Brändli and Beyne, called mod(2*n - 1).