fareylen

Returns the length of a Farey Sequence.

This opcode can be used in conjunction with GENfarey. It calculates the length of Farey Sequence F_n. Its length is given by: |F_n| = 1 + SUM over n phi(m) where phi(m) is Euler's totient function, which gives the number of integers ≤ m that are coprime to m.

Some values for the length of F_n given n:

n	F
1	2
2	3
3	5
4	7
5	11
6	13
7	19
8	23
9	29
10	33
11	43
12	47
13	59
14	65
15	73
16	81
17	97
18	103
19	121
20	129

Syntax

kfl fareylen kfn

Performance

The length of the identified Farey sequence is returned.

kfn -- Integer identifying the sequence.

Examples

Here is an example of the fareylen opcode. It uses the file fareylen.csd.

Example of the fareylen opcode.
<CsoundSynthesizer>
<CsOptions>
-n
</CsOptions>
<CsInstruments>

sr=44100
ksmps=32
nchnls=1

; by Stefano Cucchi and Menno Knevel - 2024

instr 10

kfarnf randomi 1, 10, 4
kfarn ceil kfarnf

klenght fareylen kfarn  
printks "farey length of %d = %d\n",.1,  kfarn, klenght

endin

</CsInstruments>
<CsScore>

i 10 0 3
e
</CsScore>
</CsoundSynthesizer> 

Credits

Author: Georg Boenn
University of Glamorgan, UK

New in Csound version 5.13

n	F
1	2
2	3
3	5
4	7
5	11
6	13
7	19
8	23
9	29
10	33
11	43
12	47
13	59
14	65
15	73
16	81
17	97
18	103
19	121
20	129

n	F
1	2
2	3
3	5
4	7
5	11
6	13
7	19
8	23
9	29
10	33
11	43
12	47
13	59
14	65
15	73
16	81
17	97
18	103
19	121
20	129

fareylen

Syntax

Performance

Examples

See Also

Credits

n	F
1	2
2	3
3	5
4	7
5	11
6	13
7	19
8	23
9	29
10	33
11	43
12	47
13	59
14	65
15	73
16	81
17	97
18	103
19	121
20	129