A245678 -id:A245678 - 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

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

Original entry on oeis.org

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

William Rex Marshall, Dec 16 2010

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

			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).

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. A182972 (numerators), A366191 (interleaved).
Cf. A020652, A038567, A182974-A182976.
Cf. A023022, A245677, A245678, A245718.

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

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 *)

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.

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

A245718 Integer part of sum of fractions A182972(k) / A182973(k) such that A182972(k) + A182973(k) = n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 1, 3, 1, 3, 1, 2, 1, 4, 1, 3, 2, 3, 2, 5, 1, 5, 3, 3, 2, 4, 2, 7, 3, 4, 3, 7, 2, 8, 3, 4, 3, 9, 3, 8, 3, 6, 4, 10, 3, 7, 4, 6, 5, 11, 3, 11, 5, 6, 6, 9, 4, 12, 6, 8, 4, 13, 4, 14, 6, 7, 7, 11, 4, 15, 6, 10, 7, 16

Offset: 3

Reinhard Zumkeller, Jul 30 2014

nonn

A245718(n) = floor(A245677(n)/A245678(n)).

			 See A245677.

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000

Cf. A245677, A245678, A182972, A182973.

Haskell
```
a245718 n = a245677 n `div` a245678 n
```

A245677 Numerator of sum of fractions A182972(k) / A182973(k) such that A182972(k) + A182973(k) = n.

Original entry on oeis.org

1, 1, 11, 1, 79, 26, 339, 34, 5297, 62, 69071, 1165, 11723, 9844, 471181, 2625, 8960447, 73244, 8231001, 243757, 1031626241, 151100, 4178462515, 2651758, 10396147563, 11843614, 64166447971, 362476, 1989542332021, 97275764008, 1830230212061, 57286319768

Offset: 3

Reinhard Zumkeller, Jul 30 2014

A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator;
a(n) = numerator(sum(A182972(k)/A182973(k): k such that A182972(k)+A182973(k)=n));
A245718(n) = floor(a(n)/A245678(n)).

			.     |  (num, den) = (A182973, A182973) | num(sum)| den(sum)|   [sum]
.   n |  num/den,   num + den = n        | A245677 | A245678 | A245718
. ----+----------------------------------+---------+---------+--------
.   3 |  1/2                             |       1 |       2 |       0
.   4 |  1/3                             |       1 |       3 |       0
.   5 |  1/4, 2/3                        |      11 |      12 |       0
.   6 |  1/5                             |       1 |       5 |       0
.   7 |  1/6, 2/5, 3/4                   |      79 |      60 |       1
.   8 |  1/7, 3/5                        |      26 |      35 |       0
.   9 |  1/8, 2/7, 4/5                   |     339 |     280 |       1
.  10 |  1/9, 3/7                        |      34 |      63 |       0
.  11 |  1/10, 2/9, 3/8, 4/7, 5/6        |    5297 |    2520 |       2
.  12 |  1/11, 5/7                       |      62 |      77 |       0
.  13 |  1/12, 2/11, 3/10, 4/9, 5/8, 6/7 |   69071 |   27720 |       2
.  14 |  1/13, 3/11, 5/9                 |    1165 |    1287 |       0
.  15 |  1/14, 2/13, 4/11, 7/8           |   11723 |    8008 |       1
.  16 |  1/15, 3/13, 5/11, 7/9           |    9844 |    6435 |       1 .

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.

Cf. A245678 (denominator), A182972, A182973, A245718.

import Data.Ratio ((%), numerator)
a245677 n = numerator $ sum
   [num % den | num <- [1 .. div n 2], let den = n - num, gcd num den == 1]

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

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Haskell

Mathematica

Pascal

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A245718 Integer part of sum of fractions A182972(k) / A182973(k) such that A182972(k) + A182973(k) = n.

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Haskell

A245677 Numerator of sum of fractions A182972(k) / A182973(k) such that A182972(k) + A182973(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell