A006842 - OEIS

A006842

Triangle read by rows: row n gives numerators of Farey series of order n.
(Formerly M0041)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

REFERENCES

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).

LINKS

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

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

Sequence in context: A054635 A003137 A353048 * A299038 A273693 A219967

Adjacent sequences: A006839 A006840 A006841 * A006843 A006844 A006845

KEYWORD

nonn,nice,frac,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved