A177405 -id:A177405 - cp's OEIS frontend

A177903 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 0).

0, 1, 2, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 5, 4, 4, 5, 4, 5, 6, 4, 4, 6, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 6, 7, 6, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 6, 5, 6, 7, 6, 6, 6, 6, 6, 6, 6, 5, 6, 5, 7, 7, 6, 6, 7, 7, 6, 7, 6, 6, 6, 5, 5, 7, 6, 6, 6, 7, 7, 7, 6, 6, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7

Offset: 0

N. J. A. Sloane, Dec 15 2010

nonn

Latest occurrences of odd denominators 1,3,5,7,...,29: 0,1,3,3,4,5,6,7,8,9,10,11,12,13,14,15 (The glitch in the third term reflects the fact that 2/5 and 3/5 don't show up until the 3rd iteration; whereas for n>2, it appears that the last fraction with denominator 2n+1 to show up is 1/(2n+1), and that this fraction shows up after exactly n iterations.) - James Propp

Based on postings by Richard C. Schroeppel and James Propp to the Math Fun Mailing List, Dec 15 2010.

Cf. A177405, A177407. See A178042 for another version. Cf. also A178031.

Denom[L_, k_] :=
Module[{M, i}, M = {};
  For[i = 1, i <= Length[L], i++,
   If[Denominator[L[[i]]] == k, M = Append[M, L[[i]]]]]; Return[M]]
Earliest[k_] :=
Module[{i}, For[i = 1, Length[Denom[WF[i], k]] == 0, i++]; Return[i]]
Latest[k_] :=
Module[{i}, For[i = 1, Length[Denom[WF[i], k]] < EulerPhi[k], i++];
  Return[i]]
Table[Earliest[2 n + 1], {n, 1, 100}]
(* James Propp *)

A178042 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 16 2010

nonn

Equals A177903 + 1. See that entry for further information.

A006843 Triangle read by rows: row n gives denominators of Farey series of order n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			0/1, 1/1;
0/1, 1/2, 1/1;
0/1, 1/3, 1/2, 2/3, 1/1;
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
... = A006842/A006843.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 199.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10563
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.
Andrey O. Matveev, Farey Sequences: Errata + Haskell code
N. J. A. Sloane, Stern-Brocot or Farey Tree
Eric Weisstein's World of Mathematics, Farey Sequence.
Index entries for sequences related to Stern's sequences

Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014
Row sums give A240877.
Cf. A006842 (numerators), A049455, A049456, A007305, A007306.
See also A177405/A177407.

Farey := proc(n) sort(convert(`union`({0},{seq(seq(m/k,m=1..k),k=1..n)}),list)) end: seq(denom(Farey(i)),i=1..5); # Peter Luschny, Apr 28 2009

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Denominator[ Farey[n]], {n, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[Denominator[FareySequence[n]],{n,10}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 04 2016 *)

row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k););); vf = vecsort(Set(vf)); for (i=1, #vf, print1(denominator(vf[i]), ", "));} \\ Michel Marcus, Jun 27 2014

More terms from Robert G. Wilson v, Apr 08 2004

Changed offset (=order of first row) to 1 by R. J. Mathar, Apr 26 2009

A006842 Triangle read by rows: row n gives numerators of Farey series of order n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 5, 6, 7, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5, 3, 7, 4, 5, 6, 7, 8, 1

Offset: 1

N. J. A. Sloane

			0/1, 1/1;
0/1, 1/2, 1/1;
0/1, 1/3, 1/2, 2/3, 1/1;
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
... = A006842/A006843

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923. See Vol. 1.
Guthery, Scott B. A motif of mathematics. Docent Press, 2011.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10563
Maxim Bruckheimer and Abraham Arcavi, Farey series and Pick’s area theorem, The Mathematical Intelligencer, 17.4 (1995): 64-67.
Cristian Cobeli and Alexandru Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform 5 (2003): 1-38.
Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.
Andrey O. Matveev, Farey Sequences: Errata + Haskell code
N. J. A. Sloane, Stern-Brocot or Farey Tree
Eric Weisstein's World of Mathematics, Farey Sequence.
Index entries for sequences related to Stern's sequences

Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014

Cf. A006843 (denominators), A049455, A049456, A007305, A007306. Also A177405/A177407.

Farey := proc(n) sort(convert(`union`({0},{seq(seq(m/k,m=1..k),k=1..n)}),list)) end: seq(numer(Farey(i)),i=1..5); # Peter Luschny, Apr 28 2009

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Numerator[ Farey[n]], {n, 0, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[FareySequence[n] // Numerator, {n, 1, 9}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)

row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k););); vf = vecsort(Set(vf)); for (i=1, #vf, print1(numerator(vf[i]), ", "));} \\ Michel Marcus, Jun 27 2014

More terms from Robert G. Wilson v, Apr 08 2004

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 11, 5, 17, 19, 7, 17, 13, 3, 5, 7, 9, 27, 27, 9, 7, 5, 3, 13, 17, 7, 19, 17, 5, 11, 7, 1, 1, 9, 15, 7, 25, 29, 11, 27, 21, 5, 9, 13, 17, 53, 55, 19, 15, 11, 7, 31, 41, 17, 47, 43, 13, 29, 19, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9

Offset: 0

N. J. A. Sloane, Dec 10 2010

Start with the list of fractions 0/1, 1/1 and repeatedly insert the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) between every pair of adjacent elements a/b and c/d of the list. The fractions are to be reduced before the insertion step.

			Triangle begins:
  0 1
  - -
  1 1
.
  0 1 2 1
  - - - -
  1 3 3 1
.
  0 1 2 1 4 5 2 5 4 1
  - - - - - - - - - -
  1 5 7 3 9 9 3 7 5 1

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)
Dhroova Aiylam and Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Cf. A177405, A177903, A006842/A006843.

a(44)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

cp's OEIS Frontend

A177903 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 0).

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A178042 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 1).

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Author

Keywords

Comments

A006843 Triangle read by rows: row n gives denominators of Farey series of order n.

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Author

Keywords

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References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A006842 Triangle read by rows: row n gives numerators of Farey series of order n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions