Difference between revisions of "Farey sequence"
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A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, <math>F_n</math> contains all the terms in <math>F_{n-1}</math>. Also, <math>F_n</math> contains an extra term for every number less than <math>n</math> relatively prime to <math>n</math>. Thus, we can write | A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, <math>F_n</math> contains all the terms in <math>F_{n-1}</math>. Also, <math>F_n</math> contains an extra term for every number less than <math>n</math> relatively prime to <math>n</math>. Thus, we can write | ||
| − | <math>#(F_n)=#(F_{n-1})+\phi{(n)}</math> | + | <math>\#(F_n)=\#(F_{n-1})+\phi{(n)}</math> |
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Revision as of 11:08, 9 April 2019
A Farey sequence of order 
is the sequence of all completely reduced fractions between 0 and 1 where, when in lowest terms, each fraction has a denominator less than or equal to . Each fraction starts with 0, denoted by the fraction 0/1, and ends in 1, denoted by the fraction 1/1.
Examples
Farey sequences of orders 1-4 are:
Where 
denotes a Farey sequence of order .
Properties
Sequence length
A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, contains all the terms in 
. Also, contains an extra term for every number less than relatively prime to . Thus, we can write
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