The Farey sequence
for any positive integer is the set of irreducible rational
arranged in increasing order. The first few are
A006842 and A006843). Except for ,
has an odd number of terms and the middle term is always
, and be three successive terms in a Farey series. Then
These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of
terms, insert the mediant fraction
between terms and when (Hardy and Wright 1979, pp. 25-26; Conway and
Guy 1996; Apostol 1997). Given with , let be the mediant of and . Then , and these fractions satisfy the unimodular
(Apostol 1997, p. 99).
The number of terms
in the Farey sequence for the integer is
is the totient function and is the summatory function
giving 2, 3, 5, 7, 11, 13, 19, ... (OEIS A005728).
The asymptotic limit for the function is
(Vardi 1991, p. 155).
Ford circles provide a method of visualizing the Farey sequence. The Farey sequence defines a subtree of the Stern-Brocot
tree obtained by pruning unwanted branches (Graham et al. 1994).
The Season 2 episode "
Bettor or Worse" (2006) of the television crime drama
features Farey sequences.
See also Ford Circle
Minkowski's Question Mark Function
Explore with Wolfram|Alpha
References Apostol, T. M. "Farey Fractions." §5.4 in New York: Springer-Verlag,
pp. 97-99, 1997. Modular
Functions and Dirichlet Series in Number Theory, 2nd ed. Beiler, A. H. "Farey Tails." Ch. 16
in New York:
Dover, 1966. Recreations
in the Theory of Numbers: The Queen of Mathematics Entertains. Bogomolny, A. "Farey Series, A Story." http://www.cut-the-knot.org/blue/FareyHistory.shtml. Conway,
J. H. and Guy, R. K. "Farey Fractions and Ford Circles." New York: Springer-Verlag, pp. 152-154 and 156, 1996. The
Book of Numbers. Devaney,
R. "The Mandelbrot Set and the Farey Tree, and the Fibonacci Sequence."
Amer. Math. Monthly 106, 289-302, 1999. Dickson, L. E.
Dover, pp. 155-158, 2005. History
of the Theory of Numbers, Vol. 1: Divisibility and Primality. Farey, J. "On a Curious Property
of Vulgar Fractions." London, Edinburgh and Dublin Phil. Mag. 47,
385, 1816. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Reading, MA: Addison-Wesley,
pp. 118-119, 1994. Concrete
Mathematics: A Foundation for Computer Science, 2nd ed. Guy, R. K. "Mahler's Generalization
of Farey Series." §F27 in New York: Springer-Verlag, pp. 263-265,
Problems in Number Theory, 2nd ed. Hardy, G. H. and Wright, E. M. "Farey Series and
a Theorem of Minkowski." Ch. 3 in Oxford, England: Clarendon
Press, pp. 23-37, 1979. An
Introduction to the Theory of Numbers, 5th ed. Sloane, N. J. A. Sequences A005728/M0661, A006842/M0041,
and A006843/M0081 in "The On-Line Encyclopedia
of Integer Sequences." Sylvester, J. J. "On the Number
of Fractions Contained in Any Farey Series of Which the Limiting Number is Given."
London, Edinburgh and Dublin Phil. Mag. (5th Series) 15, 251, 1883. Vardi,
I. Reading, MA: Addison-Wesley, p. 155, 1991. Computational
Recreations in Mathematica. Referenced
on Wolfram|Alpha Farey Sequence
Cite this as:
Weisstein, Eric W. "Farey Sequence." From
--A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/FareySequence.html