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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• frac
• tabf

Examples

			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.
		

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

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(denominator(vf[i]), ", "));} \\ Michel Marcus, Jun 27 2014

Extensions

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