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:

$n$ $\phi \left({n}\right)$ $m$ not coprime: $1 \le m \le n$
$1$ $1$ $\varnothing$
$2$ $1$ $2$
$3$ $2$ $3$
$4$ $2$ $2, 4$
$5$ $4$ $5$
$6$ $2$ $2, 3, 4, 6$
$7$ $6$ $7$
$8$ $4$ $2, 4, 6, 8$
$9$ $6$ $3, 6, 9$
$10$ $4$ $2, 4, 5, 6, 8, 10$

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


Euler Phi Function of $1$

$\phi \left({1}\right) = 1$


Euler Phi Function of $3$

$\phi \left({3}\right) = 2$


Euler Phi Function of $9$

The value of the Euler $\phi$ function for the integer $9$ is $6$.


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 $14$

$\phi \left({14}\right) = 6$


Euler Phi Function of $16$

The value of the Euler $\phi$ function for the integer $16$ is $8$.


Euler Phi Function of $30$

$\phi \left({30}\right) = 8$


Euler Phi Function of $42$

$\phi \left({42}\right) = 12$


Euler Phi Function of $72$

$\phi \left({72}\right) = 24$


Euler Phi Function of $78$

$\phi \left({78}\right) = 24$


Euler Phi Function of $84$

$\phi \left({84}\right) = 24$


Euler Phi Function of $87$

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


Euler Phi Function of $90$

$\phi \left({90}\right) = 24$


Euler Phi Function of $216$

$\phi \left({216}\right) = 72$


Euler Phi Function of $222$

$\phi \left({222}\right) = 72$


Euler Phi Function of $228$

$\phi \left({228}\right) = 72$


Euler Phi Function of $234$

$\phi \left({234}\right) = 72$


Euler Phi Function of $248$

$\phi \left({248}\right) = 120$


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

The value of the Euler $\phi$ function for the integer $1\,000\,000$ is $400\,000$.


Successive Solutions of $\phi (n) = \phi (n + 2)$

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

$\phi \left({n}\right) = \phi \left({n + 2}\right)$


Sources