Investigating the Phi Function
The phi function is defined for any positive integer[IMAGE](n), as the
number of positive integers not greater than and co-prime (have no
factor other than 1 in common) to n
Example
So [IMAGE](12) = 4 because the integers less than 12 which have no
factors in common with it except for 1 are 1,5,7,11 i.e. there is 4 of
them.
I started to investigate the phi function of numbers from 2 to 24 so
I could find patterns, which I can use to create a formula for the[IMAGE](n)
term
[IMAGE](n)
Shared factors
Not sharing factors
[IMAGE] (2)
-
1
[IMAGE] (2) = 1
[IMAGE](3)
1,2
[IMAGE](3) = 2
[IMAGE](4)
2
1,3
[IMAGE](4) = 2
[IMAGE](5)
1,2,3,4
[IMAGE](5) = 4
[IMAGE](6)
2,3,4
1,5
[IMAGE](6) = 2
[IMAGE](7)
1,2,3,4,5,6
[IMAGE](7) = 6
[IMAGE](8)
2,4,6
1,3,5,7
[IMAGE](8) = 4
[IMAGE](9)
3,6
1,2,4,5,7,8
[IMAGE](9) = 6
[IMAGE](10)
2,4,6,8,5
1,3,7,9
[IMAGE](10) = 4
[IMAGE](11)
1,2,3,4,5,6,7,8,9,10
[IMAGE](11) = 10
[IMAGE](12)
2,4,6,8,10,3,9
1,5,7,11
[IMAGE](12) = 4
[IMAGE](13)
1,2,3,4,5,6,7,8,9,10,11,12,
[IMAGE](13) = 12
[IMAGE](14)
2,4,6,8,10,12,7
1,3,5,11,13
[IMAGE](14) = 6
[IMAGE](15)
3,5,9,12,6,10
1,2,4,7,8,11,13,14
[IMAGE](15) = 8
[IMAGE](16)
2,4,6,8,10,12,14
1,3,5,7,11,13,15
[IMAGE](16) = 8
[IMAGE](17)
1,2,3,4,5,6,7,8,9,10,11,12,13,
14,15,16
[IMAGE](17) = 16
[IMAGE](18)
2,3,4,6,8,10,12,14
1,5,7,11,13,17,
[IMAGE](18) = 6
[IMAGE](19)
1,2,3,4,5,6,7,8,910,11,12,13,14,
15,16,17,18
[IMAGE](19) = 18
[IMAGE](20)
2,4,6,8,10,12,1,4,16,18,5,15
1,2,4,5,8,10,11,13,16,17,19
[IMAGE](20) = 8
[IMAGE](21)
3,6,9,12,15,18,7,14
1,2,4,5,8,10,11,13,16,17,19,20
[IMAGE](21) = 12
[IMAGE](22)
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