Investigating the Phi Function

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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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