A182973 Denominators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

2, 3, 4, 3, 5, 6, 5, 4, 7, 5, 8, 7, 5, 9, 7, 10, 9, 8, 7, 6, 11, 7, 12, 11, 10, 9, 8, 7, 13, 11, 9, 14, 13, 11, 8, 15, 13, 11, 9, 16, 15, 14, 13, 12, 11, 10, 9, 17, 13, 11, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 17, 13, 11, 20, 19, 17, 16, 13, 11, 21, 19, 17, 15, 13, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12

Offset: 1

Author

William Rex Marshall, Dec 16 2010

Keywords

• nonn
• easy
• frac
• nice

Comments

A023022(n) and A245678(n) give number and denominator of sum of fractions A182972(k)/a(k) such that A182972(k) + a(k) = n. - Reinhard Zumkeller, Jul 30 2014

Examples

			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, ...
(this is A182972/A182973).
		

References

• 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.

Links

• 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

Crossrefs

Cf. A182972 (numerators), A366191 (interleaved).

Cf. A020652, A038567, A182974-A182976.

Cf. A023022, A245677, A245678, A245718.

Programs

Haskell

a182973 n = a182973_list !! (n-1)
a182973_list = map snd $ 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

Mathematica

A182973list[s_] := Table[If[CoprimeQ[num, s-num], s-num, Nothing], {num, Floor[s/2]}]; Flatten[Array[A182973list, 25, 3]] (* Paolo Xausa, Feb 27 2024 *)

Pascal

program a182973;
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,' ',den);
  until n=100000;
end.

Python

from itertools import count, islice
from math import gcd
def A182973_gen(): # generator of terms
    return (n-i for n in count(2) for i in range(1,1+(n-1>>1)) if gcd(i,n-i)==1)
A182973_list = list(islice(A182973_gen(),10)) # Chai Wah Wu, Aug 28 2023