Euler Phi Function/Examples

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Examples of Euler $\phi$ Function

The values of the Euler $\phi$ function for the first few integers are as follows:

$n$ $\map \phi n$ $m$ not coprime: $1 \le m \le n$
$1$ $1$ $\O$
$2$ $1$ $2$
$3$ $2$ $3$
$4$ $2$ $2, 4$
$5$ $4$ $5$
$6$ $2$ $2, 3, 4, 6$
$7$ $6$ $7$
$8$ $4$ $2, 4, 6, 8$
$9$ $6$ $3, 6, 9$
$10$ $4$ $2, 4, 5, 6, 8, 10$

This sequence is A000010 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Euler Phi Function of $1$

$\map \phi 1 = 1$


Euler Phi Function of $2$

$\map \phi 2 = 1$


Euler Phi Function of $3$

$\map \phi 3 = 2$


Euler Phi Function of $4$

$\map \phi 4 = 2$


Euler Phi Function of $9$

$\map \phi 9 = 6$


Numbers for which Euler Phi Function is $6$

There are $4$ numbers for which the value of the Euler $\phi$ function is $6$:

$7, 9, 14, 18$


Euler Phi Function of $14$

$\map \phi {14} = 6$


Euler Phi Function of $16$

$\map \phi {16} = 8$


Euler Phi Function of $20$

$\map \phi {20} = 8$


Euler Phi Function of $24$

$\map \phi {24} = 8$


Euler Phi Function of $30$

$\map \phi {30} = 8$


Euler Phi Function of $42$

$\map \phi {42} = 12$


Euler Phi Function of $72$

$\map \phi {72} = 24$


Euler Phi Function of $78$

$\map \phi {78} = 24$


Euler Phi Function of $84$

$\map \phi {84} = 24$


Euler Phi Function of $87$

$\phi \left({87}\right) = 56$


Euler Phi Function of $90$

$\map \phi {90} = 24$


Euler Phi Function of $216$

$\map \phi {216} = 72$


Euler Phi Function of $222$

$\map \phi {222} = 72$


Euler Phi Function of $228$

$\map \phi {228} = 72$


Euler Phi Function of $234$

$\map \phi {234} = 72$


Euler Phi Function of $248$

$\map \phi {248} = 120$


Euler Phi Function of $1\,000\,000$

$\map \phi {1 \, 000 \, 000} = 400 \, 000$


Successive Solutions of $\map \phi n = \map \phi {n + 2}$

$7$ and $8$ are two successive integers which are solutions to the equation:

$\map \phi n = \map \phi {n + 2}$


Table of Values of Euler $\phi$ Function

The Euler $\phi$ function for the first $100$ positive integers is as follows:


$\begin{array} {|r|r|}

\hline n & \map \phi n \\ \hline 1 & 1 \\ 2 & 1 \\ 3 & 2 \\ 4 & 2 \\ 5 & 4 \\ 6 & 2 \\ 7 & 6 \\ 8 & 4 \\ 9 & 6 \\ 10 & 4 \\ 11 & 10 \\ 12 & 4 \\ 13 & 12 \\ 14 & 6 \\ 15 & 8 \\ 16 & 8 \\ 17 & 16 \\ 18 & 6 \\ 19 & 18 \\ 20 & 8 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \phi n \\ \hline 21 & 12 \\ 22 & 10 \\ 23 & 22 \\ 24 & 8 \\ 25 & 20 \\ 26 & 12 \\ 27 & 18 \\ 28 & 12 \\ 29 & 28 \\ 30 & 8 \\ 31 & 30 \\ 32 & 16 \\ 33 & 20 \\ 34 & 16 \\ 35 & 24 \\ 36 & 12 \\ 37 & 36 \\ 38 & 18 \\ 39 & 24 \\ 40 & 16 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \phi n \\ \hline 41 & 40 \\ 42 & 12 \\ 43 & 42 \\ 44 & 20 \\ 45 & 24 \\ 46 & 22 \\ 47 & 46 \\ 48 & 16 \\ 49 & 42 \\ 50 & 20 \\ 51 & 32 \\ 52 & 24 \\ 53 & 52 \\ 54 & 18 \\ 55 & 40 \\ 56 & 24 \\ 57 & 36 \\ 58 & 28 \\ 59 & 58 \\ 60 & 16 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \phi n \\ \hline 61 & 60 \\ 62 & 30 \\ 63 & 36 \\ 64 & 32 \\ 65 & 48 \\ 66 & 20 \\ 67 & 66 \\ 68 & 32 \\ 69 & 44 \\ 70 & 24 \\ 71 & 70 \\ 72 & 24 \\ 73 & 72 \\ 74 & 36 \\ 75 & 40 \\ 76 & 36 \\ 77 & 60 \\ 78 & 24 \\ 79 & 78 \\ 80 & 32 \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline n & \map \phi n \\ \hline 81 & 54 \\ 82 & 40 \\ 83 & 82 \\ 84 & 24 \\ 85 & 64 \\ 86 & 42 \\ 87 & 56 \\ 88 & 40 \\ 89 & 88 \\ 90 & 24 \\ 91 & 72 \\ 92 & 44 \\ 93 & 60 \\ 94 & 46 \\ 95 & 72 \\ 96 & 32 \\ 97 & 96 \\ 98 & 42 \\ 99 & 60 \\ 100 & 40 \\ \hline \end{array}$


Sources

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