Euler Phi Function/Examples
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:
$\quad \begin{array} {|r|r|}
\hline
n & \map \phi n \\
\hline
1 & 1 \\
2 & 1 \\
3 & 2 \\
4 & 2 \\
5 & 4 \\
6 & 2 \\
7 & 6 \\
8 & 4 \\
9 & 6 \\
10 & 4 \\
11 & 10 \\
12 & 4 \\
13 & 12 \\
14 & 6 \\
15 & 8 \\
16 & 8 \\
17 & 16 \\
18 & 6 \\
19 & 18 \\
20 & 8 \\
\hline
\end{array} \qquad
\begin{array} {|r|r|}
\hline
n & \map \phi n \\
\hline
21 & 12 \\
22 & 10 \\
23 & 22 \\
24 & 8 \\
25 & 20 \\
26 & 12 \\
27 & 18 \\
28 & 12 \\
29 & 28 \\
30 & 8 \\
31 & 30 \\
32 & 16 \\
33 & 20 \\
34 & 16 \\
35 & 24 \\
36 & 12 \\
37 & 36 \\
38 & 18 \\
39 & 24 \\
40 & 16 \\
\hline
\end{array} \qquad
\begin{array} {|r|r|}
\hline
n & \map \phi n \\
\hline
41 & 40 \\
42 & 12 \\
43 & 42 \\
44 & 20 \\
45 & 24 \\
46 & 22 \\
47 & 46 \\
48 & 16 \\
49 & 42 \\
50 & 20 \\
51 & 32 \\
52 & 24 \\
53 & 52 \\
54 & 18 \\
55 & 40 \\
56 & 24 \\
57 & 36 \\
58 & 28 \\
59 & 58 \\
60 & 16 \\
\hline
\end{array} \qquad
\begin{array} {|r|r|}
\hline
n & \map \phi n \\
\hline
61 & 60 \\
62 & 30 \\
63 & 36 \\
64 & 32 \\
65 & 48 \\
66 & 20 \\
67 & 66 \\
68 & 32 \\
69 & 44 \\
70 & 24 \\
71 & 70 \\
72 & 24 \\
73 & 72 \\
74 & 36 \\
75 & 40 \\
76 & 36 \\
77 & 60 \\
78 & 24 \\
79 & 78 \\
80 & 32 \\
\hline
\end{array} \qquad
\begin{array} {|r|r|}
\hline
n & \map \phi n \\
\hline
81 & 54 \\
82 & 40 \\
83 & 82 \\
84 & 24 \\
85 & 64 \\
86 & 42 \\
87 & 56 \\
88 & 40 \\
89 & 88 \\
90 & 24 \\
91 & 72 \\
92 & 44 \\
93 & 60 \\
94 & 46 \\
95 & 72 \\
96 & 32 \\
97 & 96 \\
98 & 42 \\
99 & 60 \\
100 & 40 \\
\hline
\end{array}$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $27$