cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A182972 Numerators of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator.

Original entry on oeis.org

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

Views

Author

William Rex Marshall, Dec 16 2010

Keywords

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

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
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).

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

Crossrefs

Cf. A182973 (denominators), A366191 (interleaved).
Cf. A020652, A038567, A182974, A182975, A182976.
Cf. A023022, A245677, A245678, A245718.
Essentially the same as A333856.

Programs

  • 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
  • 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 *)
  • 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.
  • 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

Extensions

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

Views

Author

William Rex Marshall, Dec 16 2010

Keywords

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

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

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

Original entry on oeis.org

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

Views

Author

Sam Alexander, Feb 27 2004

Keywords

Comments

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

Examples

			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;

References

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

Links

Crossrefs

Cf. A092543, A056011, A056023.
Variants of Cantor's enumeration are: A352911, A366191, A319571, A354266.

Programs

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

Formula

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

Extensions

Name edited by Michel Marcus, Dec 14 2023

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

Views

Author

N. J. A. Sloane, Apr 09 2022

Keywords

Comments

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

Examples

			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],
  ...

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    A352911row[n_]:=Select[Array[{#,n-#}&,n-1],CoprimeQ[First[#],Last[#]]&];
Array[A352911row,10,2] (* Generates 10 rows *) (* Paolo Xausa, Oct 10 2023 *)
  • Python
    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
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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