A020652 -id:A020652 - 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

A038569 Denominators in a certain bijection from positive integers to positive rationals.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

See A020652/A020653 for an alternative version where the fractions p/q are listed by increasing p+q, then p. - M. F. Hasler, Nov 25 2021

			First arrange the positive fractions p/q <= 1 by increasing denominator, then by increasing numerator:
1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567).
Now follow each but the first term by its reciprocal:
1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, ... (this is A038568/A038569).

H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

Cf. A038566, A038567, A038568.

See A020652, A020653 for an alternative version.

with (numtheory): A038569 := proc (n) local sum, j, k; sum := 1: k := 2: while (sum < n) do: sum := sum + 2 * phi(k): k := k + 1: od: sum := sum - 2 * phi(k-1): j := 1: while sum < n do: if gcd(j,k-1) = 1 then sum := sum + 2: fi: j := j+1: od: if sum > n then RETURN (k-1) fi: RETURN (j-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

a[n_] := Module[{s = 1, k = 2, j = 1}, While[s <= n, s = s + 2*EulerPhi[k]; k = k+1]; s = s - 2*EulerPhi[k-1]; While[s <= n, If[GCD[j, k-1] == 1, s = s+2]; j = j+1]; If[s > n+1, k-1, j-1]]; Table[a[n], {n, 0, 99}](* Jean-François Alcover, Nov 10 2011, after Maple *)

a(n) = { my (e); for (q=1, oo, if (n+1<2*e=eulerphi(q), for (p=1, oo, if (gcd(p,q)==1, if (n+1<2, return ([q,p][n+2]), n-=2))), n-=2*e)) } \\ Rémy Sigrist, Feb 25 2021

from sympy import totient, gcd
def a(n):
    s=1
    k=2
    while s<=n:
        s+=2*totient(k)
        k+=1
    s-=2*totient(k - 1)
    j=1
    while s<=n:
        if gcd(j, k - 1)==1: s+=2
        j+=1
    if s>n + 1: return k - 1
    return j - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica

More terms from Erich Friedman

Definition clarified by N. J. A. Sloane, Nov 25 2021

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

A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 19 2002

The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.

			The Sagher map sends the following fractions to 1, 2, 3, 4, ...: 1/1, 1/2, 1/3, 2/1, 1/5, 1/6, 1/7, 1/4, 3/1, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025 [math.HO], 2005.
Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, pp. 1013-1014.
Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p. 868.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
Index entries for sequences related to enumerating the rationals

Cf. A071975. Differs from A056622 at a(32).
Cf. A000290, A027748, A077591, A124010, A350388.
For other bijective mappings from integers to positive rationals see A002487, A020652/A020653, A038568/A038569, A229994/A077610, A295515.
Cf. A307868.

a071974 n = product $ zipWith (^) (a027748_row n) $
   map (\e -> (1 - e `mod` 2) * e `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012

f[{p_, a_}] := If[EvenQ[a], p^(a/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)

a(n)=local(v=factor(n)~); prod(k=1,length(v),if(v[2,k]%2,1,v[1,k]^(v[2,k]/2)))

from math import prod
from sympy import factorint
def A071974(n): return prod(p**(e>>1) for p, e in factorint(n).items() if e&1^1) # Chai Wah Wu, Jul 27 2024

If n=Product p_i^e_i, then a_n=Product p_i^f(e_i), where f(n)=n/2 if n is even and f(n)=0 if n is odd. - Reiner Martin, Jul 08 2002
a(n^2) = n, A071975(n^2) = 1, cf. A000290; a(2*(2*n-1)^2) = 2*n+1, A071975(2*(2*n-1)^2) = 2, cf. A077591. - Reinhard Zumkeller, Jul 10 2011
From Amiram Eldar, Nov 02 2023, Jul 26 2024: (Start)
a(n) = sqrt(A350388(n)) (square root of largest unitary divisor of n that is a square).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-1)). (End)
From Vaclav Kotesovec, May 05 2025: (Start)
Let f(s) = Product_{p prime} (1 -  (p^s + p)/((p^s + 1)*p^(2*s))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(1+p))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321...
f'(1) = f(1) * Sum_{p prime} (5*p+3)*log(p) / ((p+1)*(p^2+p-2)) = f(1) * 2.1244279471327068377850377690765768532203174482128717024402373817115555...
and gamma is the Euler-Mascheroni constant A001620. (End)

More terms from Reiner Martin, Jul 08 2002

Additional references supplied by Kevin Ryde added by N. J. A. Sloane, May 31 2012

A038568 Numerators in canonical bijection from positive integers to positive rationals.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Even-indexed terms are positive integers in order, with m occurring phi(m) times. Preceding odd-indexed terms (except for missing initial 0) are the corresponding numbers <= m and relatively prime to m, in increasing order. The denominators are just this sequence shifted left. Thus each positive rational occurs exactly once as a ratio a(n)/a(n+1). - Franklin T. Adams-Watters, Dec 06 2006

			First arrange fractions by increasing denominator, then by increasing numerator:
1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567);
now follow each term (except the first) with its reciprocal:
1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, ... (this is A038568/A038569).

H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

Cf. A020652, A020653, A038566, A038567, A038569.

using Nemo
function A038568List(len)
    a, A = QQ(0), []
    for n in 1:len
        a = next_minimal(a)
        push!(A, numerator(a))
    end
A end
A038568List(84) |> println # Peter Luschny, Mar 13 2018

with (numtheory): A038568 := proc (n) local sum, j, k; sum := 1: k := 2: while (sum < n) do: sum := sum + 2 * phi(k): k := k + 1: od: sum := sum - 2 * phi(k-1): j := 1: while sum < n do: if gcd(j,k-1) = 1 then sum := sum + 2: fi: j := j+1: od: if sum > n then RETURN (j-1) fi: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

a[n_] := Module[{sum = 1, k = 2}, While[sum < n, sum = sum + 2*EulerPhi[k]; k = k+1]; sum = sum - 2*EulerPhi[k-1]; j = 1; While[sum < n, If[GCD[j, k-1] == 1, sum = sum+2]; j = j+1; ]; If[sum > n, Return[j-1]]; Return[k-1] ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 21 2012, translated from Maple *)

a(n) = { my (e); for (q=1, oo, if (n+1<2*e=eulerphi(q), for (p=1, oo, if (gcd(p,q)==1, if (n+1<2, return ([p,q][n+2]), n-=2))), n-=2*e)) } \\ Rémy Sigrist, Feb 25 2021

from sympy import totient, gcd
def a(n):
    s=1
    k=2
    while sn: return j - 1
    return k - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica

More terms from Erich Friedman

A048705 The rule numbers for 1-D CA composed of Rules "90" and "150" so that each direction occurs only once.

Original entry on oeis.org

90, 150, 1721342310, 140117185019831836588493434554119984790, 113427455640312821160607117168492587690

Offset: 1

Antti Karttunen, Mar 09 1999

nonn

The "numerator" (0, 1 and the rest from A020652) is the multiplicity of the "Rule 150" component and the "denominator" (1, 0 and the rest from A020653) is the multiplicity of the "Rule 90" component.
The resulting numbers define one-dimensional linear cellular automata with radius being the sum of the number of the "90" and "150" components.
In hexadecimal the sequence is 5A, 96, 66999966, 69699696969669699696696969699696, 5555555555555555AAAAAAAAAAAAAAAA, ...

Index entries for sequences related to cellular automata

A048706 gives the corresponding "XOR-conjugate" rules.

Cf. A038183, A038184, A048709 (for specific examples). See also A048708, A048720.

# The definitions of bit_i and floor_log_2 are given in A048700
rule90 := proc(seed,n) option remember: local sl, i: if (0 = n) then (seed) else sl := floor_log_2(seed+1); add(((bit_i(rule90(seed,n-1),i)+bit_i(rule90(seed,n-1),i-2)) mod 2)*(2^i), i=0..(2*n)+sl) fi: end:
rule150 := proc(seed,n) option remember: local sl, i: if (0 = n) then (seed) else sl := floor_log_2(seed+1);
add(((bit_i(rule150(seed,n-1),i)+bit_i(rule150(seed,n-1),i-1)+bit_i(rule150(seed,n-1),i-2)) mod 2)*(2^i), i=0..((2*n)+sl)) fi: end:
# Rule 90 and Rule 150 are commutative in respect to each other:
rule90x150combination := proc(n) local p,q,i; p := extended_A020652[ n ]; # the Rule 150 component [ 0,1,op(A020652) ]
q := extended_A020653[ n ]; # the Rule 90 component [ 1,0,op(A020653) ]
RETURN(sum('bit_i(rule150(rule90(i,q),p),(2*(p+q))) * (2^i)','i'=0..(2^((2*(p+q))+1))-1));
end:

a(n) = rule90x150combination(n) # See the Maple procedures below.

A352911 Cantor's List: Pairs (i, j) of relatively prime positive integers sorted first by i + j then by i.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2022

a(2*n-1) / a(2*n) is the n-th fraction in Cantor's enumeration of the positive rational numbers. - Peter Luschny, Oct 10 2023

			The first few pairs are, seen as an irregular triangle:
  [1, 1],
  [1, 2], [2, 1],
  [1, 3], [3, 1],
  [1, 4], [2, 3], [3, 2], [4, 1],
  [1, 5], [5, 1],
  [1, 6], [2, 5], [3, 4], [4, 3], [5, 2], [6, 1],
  [1, 7], [3, 5], [5, 3], [7, 1],
  [1, 8], [2, 7], [4, 5], [5, 4], [7, 2], [8, 1],
  [1, 9], [3, 7], [7, 3], [9, 1],
  ...

N. J. A. Sloane, Table of n, a(n) for n = 1..9914
Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258, (p. 250).
N. J. A. Sloane, List of the 4957 pairs (i,j) with i+j <= 127. [Note this is not a b-file.]
Index entries for sequences related to enumerating the rationals

Cf. A352909, A020652 or A038566 (i-coordinates), A020653 (j-coordinates), A366191.

CantorsList := proc(upto) local C, F, n, t, count;
C := NULL; count := 0:
for n from 2 while count < upto do
    F := select(t -> igcd(t, n-t) = 1, [$1..n-1]);
    C := C, seq([t, n - t], t = F);
    count := count + nops(F) od:
ListTools:-Flatten([C]) end:
CantorsList(40);  # Peter Luschny, Oct 10 2023

A352911row[n_]:=Select[Array[{#,n-#}&,n-1],CoprimeQ[First[#],Last[#]]&];
Array[A352911row,10,2] (* Generates 10 rows *) (* Paolo Xausa, Oct 10 2023 *)

from math import gcd
from itertools import chain, count, islice
def A352911_gen(): # generator of terms
    return chain.from_iterable((i,n-i) for n in count(2) for i in range(1,n) if gcd(i,n-i)==1)
A352911_list = list(islice(A352911_gen(),30)) # Chai Wah Wu, Oct 10 2023

A157807 Numerators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

Original entry on oeis.org

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

Offset: 1

Ron R. King, Mar 07 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A157813 (denominators), A038566.

With Cantor's ordering: A020652, A020653, A352911.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
  S:= select(t -> igcd(t,m-t)=1, [$1..m-1]);
  count:= count+nops(S);
  if m::even then R:= R, op(S) else R:= R, seq(m-t,t=S) fi;
od:
R; # Robert Israel, Oct 09 2023

from math import gcd
for s in range(2, 100, 2):
  for i in range(1, s):
    if gcd(i, s - i) != 1: continue
    print(i)
  for i in range(s, 0, -1):
    if gcd(i, s + 1 - i) != 1: continue
    print(i)
# Hiroaki Yamanouchi, Oct 06 2014

A-number in cross-reference corrected by R. J. Mathar, Sep 23 2009
a(19)-a(20) corrected and a(58)-a(82) added by Hiroaki Yamanouchi, Oct 06 2014
Name corrected by Andrey Zabolotskiy, Oct 10 2023

A157813 Denominators of fractions arranged in "antidiagonal boustrophedon" ordering with equivalent fractions removed: (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, 5/2, ...).

Original entry on oeis.org

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

Offset: 1

Ron R. King, Mar 07 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A157807 (numerators), A038567.

With Cantor's ordering: A020652, A020653, A352911.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
  S:= select(t -> igcd(t,m-t)=1, [$1..m-1]);
  count:= count+nops(S);
  if m::odd then R:= R, op(S) else R:= R, seq(m-t,t=S) fi;
od:
R; # Robert Israel, Oct 09 2023

from math import gcd
for s in range(2, 100, 2):
  for i in range(1, s):
    if gcd(i, s - i) != 1: continue
    print(s - i)
  for i in range(s, 0, -1):
    if gcd(i, s + 1 - i) != 1: continue
    print(s + 1 - i)
# Hiroaki Yamanouchi, Oct 06 2014

a(58)-a(83) from Hiroaki Yamanouchi, Oct 06 2014

Name corrected by Andrey Zabolotskiy, Oct 10 2023

A037161 Well-order the rational numbers; take numerators.

Original entry on oeis.org

0, -1, 1, -2, -1, 1, 2, -3, -1, 1, 3, -4, -3, -2, -1, 1, 2, 3, 4, -5, -1, 1, 5, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, -7, -5, -3, -1, 1, 3, 5, 7, -8, -7, -5, -4, -2, -1, 1, 2, 4, 5, 7, 8, -9, -7, -3, -1, 1, 3, 7, 9, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1

Offset: 0

N. J. A. Sloane

W. Sierpiński, Cardinal and Ordinal Numbers, Warsaw 1965, 2nd ed., p. 40.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A037162.

Cf. A020652.

import Data.List (transpose)
import Data.Ratio ((%), numerator)
a037161 n = a037161_list !! n
a037161_list = 0 : map numerator
  (concat $ concat $ transpose [map (map negate) qss, map reverse qss])
  where qss = map q [1..]
        q x = map (uncurry (%)) $ filter ((== 1) . uncurry gcd) $
                  zip (reverse zs) zs where zs = [1..x]
-- Reinhard Zumkeller, Mar 08 2013

order[n_] := Join[-Reverse[ pos = Select[(r = Range[n])/Reverse[r], Numerator[#] + Denominator[#] == n + 1 & ] ], pos]; order[0] = 0; Numerator[ Flatten[ Table[ order[n], {n, 0, 10}]]] (* Jean-François Alcover, Jun 27 2012 *)

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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A038569 Denominators in a certain bijection from positive integers to positive rationals.

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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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A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

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A038568 Numerators in canonical bijection from positive integers to positive rationals.

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Julia

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A048705 The rule numbers for 1-D CA composed of Rules "90" and "150" so that each direction occurs only once.

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