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

1,4

Comments

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

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.

Example

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).

Maple

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;

Mathematica

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 *)

Prog

(Pascal) 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.
(Haskell)
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
(Python)
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

Crossrefs

Cf. A182973 (denominators), A366191 (interleaved).

Cf. A020652, A038567, A182974, A182975, A182976.

Cf. A023022, A245677, A245678, A245718.

Essentially the same as A333856.

Sequence in context: A083796 A037039 A333856 * A153452 A090680 A133771

Adjacent sequences: A182969 A182970 A182971 * A182973 A182974 A182975

Keyword

nonn,easy,frac,nice

Author

William Rex Marshall, Dec 16 2010

Extensions

Corrected by William Rex Marshall, Aug 12 2013

Status

approved