A352911 -id:A352911 - cp's OEIS frontend

A092542 Table whose n-th row is constant and equal to n, read by antidiagonals alternately upwards and downwards.

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

Offset: 1

Sam Alexander, Feb 27 2004

Let A be sequence A092542 (this sequence) and B be sequence A092543 (1, 2, 1, 1, 2, 3, 4, ...). Under upper trimming or lower trimming, A transforms into B and B transforms into A. Also, B gives the number of times each element of A appears. For example, A(7) = 1 and B(7) = 4 because the 1 in A(7) is the fourth 1 to appear in A. - Kerry Mitchell, Dec 28 2005
First inverse function (numbers of rows) for pairing function A056023 and second inverse function (numbers of columns) for pairing function A056011. - Boris Putievskiy, Dec 24 2012
The rational numbers a(n)/A092543(n) can be systematically ordered and numbered in this way, as Georg Cantor first proved in 1873. - Martin Renner, Jun 05 2016

			The table
  1 1 1 1 1 ...
  2 2 2 2 2 ...
  3 3 3 3 3 ...
  4 4 4 4 4 ...
gives
  1;
  1 2;
  3 2 1;
  1 2 3 4;
  5 4 3 2 1;
  1 2 3 4 5 6;

Amir D. Aczel, "The Mystery of the Aleph, Mathematics, the Kabbalah and the Search for Infinity", Barnes & Noble, NY 2000, page 112.

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's MathWorld, Pairing functions

Cf. A092543, A056011, A056023.

Variants of Cantor's enumeration are: A352911, A366191, A319571, A354266.

Table[ Join[Range[2n - 1], Reverse@ Range[2n - 2]], {n, 8}] // Flatten (* Robert G. Wilson v, Sep 28 2006 *)

a(n) = ((-1)^t+1)*j/2-((-1)^t-1)*i/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

Name edited by Michel Marcus, Dec 14 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

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 *)

A366192 Pairs (i, j) of noncoprime positive integers sorted first by i + j then by i.

Original entry on oeis.org

2, 2, 2, 4, 3, 3, 4, 2, 2, 6, 4, 4, 6, 2, 3, 6, 6, 3, 2, 8, 4, 6, 5, 5, 6, 4, 8, 2, 2, 10, 3, 9, 4, 8, 6, 6, 8, 4, 9, 3, 10, 2, 2, 12, 4, 10, 6, 8, 7, 7, 8, 6, 10, 4, 12, 2, 3, 12, 5, 10, 6, 9, 9, 6, 10, 5, 12, 3, 2, 14, 4, 12, 6, 10, 8, 8, 10, 6, 12, 4, 14, 2

Offset: 1

Peter Luschny, Oct 10 2023

The rows of A290600 interleaved term by term with the reversed rows of A290600. - Peter Munn, Jan 28 2024

			The first few pairs are, seen as an irregular triangle (where rows with a prime index are empty (and are therefore missing)):
  [2,  2],
  [2,  4], [3,  3], [4, 2],
  [2,  6], [4,  4], [6, 2],
  [3,  6], [6,  3],
  [2,  8], [4,  6], [5, 5], [6, 4], [ 8, 2],
  [2, 10], [3,  9], [4, 8], [6, 6], [ 8, 4], [ 9, 3], [10, 2],
  [2, 12], [4, 10], [6, 8], [7, 7], [ 8, 6], [10, 4], [12, 2],
  [3, 12], [5, 10], [6, 9], [9, 6], [10, 5], [12, 3],
  ...
There are A016035(n) pairs in row n.

Paolo Xausa, Table of n, a(n) for n = 1..9886

Cf. A016035, A290600 (first bisection), A352911 (complement).

aList := proc(upto) local F, P, n, t, count;
P := NULL; count := 0:
for n from 2 while count < upto do
    F := select(t -> igcd(t, n - t) <> 1, [$1..n-1]);
    P := P, seq([t, n - t], t = F);
    count := count + nops([F]) od:
ListTools:-Flatten([P]) end:
aList(16);

A366192row[n_]:=Select[Array[{#,n-#}&,n-1],!CoprimeQ[First[#],Last[#]]&];
Array[A366192row,20,2] (* Paolo Xausa, Nov 28 2023 *)

from math import gcd
from itertools import chain, count, islice
def A366192_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)
A366192_list = list(islice(A366192_gen(),30)) # Chai Wah Wu, Oct 10 2023

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A092542 Table whose n-th row is constant and equal to n, read by antidiagonals alternately upwards and downwards.

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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, ...).

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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, ...).

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

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Programs

Maple

Mathematica

A366192 Pairs (i, j) of noncoprime positive integers sorted first by i + j then by i.

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