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A006843 Triangle read by rows: row n gives denominators of Farey series of order n.
(Formerly M0081)
31
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152

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.

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

Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981, 2008

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(denom(Farey(i)), i=1..5); # [From 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 *)

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

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.

Sequence in context: A132463 A153901 A132844 * A049456 A117506 A179205

Adjacent sequences:  A006840 A006841 A006842 * A006844 A006845 A006846

KEYWORD

nonn,nice,frac,tabf

AUTHOR

N. J. A. Sloane.

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

STATUS

approved

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Last modified December 8 01:14 EST 2016. Contains 278902 sequences.