Euler function

[logistic_guy]

\(\displaystyle \varphi(17) = 17^0(17 - 1) = \textcolor{red}{\bold{16}}\)

 

[logistic_guy]

\(\displaystyle \varphi(18) = \varphi(2 \times 9) = \varphi(2)\varphi(3^2) = 2^0(2 - 1)3^1(3 - 1) = \textcolor{purple}{\bold{6}}\)

 

[logistic_guy]

\(\displaystyle \varphi(19) = 19^0(19 - 1) = \textcolor{red}{\bold{18}}\)

 

[logistic_guy]

\(\displaystyle \varphi(20) = \varphi(4 \times 5) = \varphi(2^2)\varphi(5) = 2^1(2 - 1)5^0(5 - 1) = \textcolor{blue}{\bold{8}}\)

 

[logistic_guy]

\(\displaystyle \varphi(21) = \varphi(3 \times 7) = \varphi(3)\varphi(7) = 3^0(3 - 1)7^0(7 - 1) = \textcolor{green}{\bold{12}}\)

 

[logistic_guy]

\(\displaystyle \varphi(22) = \varphi(2 \times 11) = \varphi(2)\varphi(11) = 2^0(2 - 1)11^0(11 - 1) = \textcolor{darkblue}{\bold{10}}\)

 

[logistic_guy]

\(\displaystyle \varphi(23) = 23^0(23 - 1) = \textcolor{darkred}{\bold{22}}\)

 

[logistic_guy]

\(\displaystyle \varphi(24) = \varphi(8 \times 3) = \varphi(2^3)\varphi(3) = 2^2(2 - 1)3^0(3 - 1) = \textcolor{blue}{\bold{8}}\)

 

[logistic_guy]

\(\displaystyle \varphi(25) = \varphi(5^2) = 5^1(5 - 1) = \textcolor{red}{\bold{20}}\)

 

[logistic_guy]

\(\displaystyle \varphi(26) = \varphi(2 \times 13) = \varphi(2)\varphi(13) = 2^0(2 - 1)13^0(13 - 1) = \textcolor{green}{\bold{12}}\)

 

[logistic_guy]

\(\displaystyle \varphi(27) = \varphi(3^3) = 3^2(3 - 1) = \textcolor{purple}{\bold{18}}\)

 

[logistic_guy]

\(\displaystyle \varphi(28) = \varphi(4 \times 7) = \varphi(2^2)\varphi(7) = 2^1(2 - 1)7^0(7 - 1) = \textcolor{blue}{\bold{12}}\)

 

[logistic_guy]

\(\displaystyle \varphi(29) = 29^0(29 - 1) = \textcolor{red}{\bold{28}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{FINALLY}}\)

\(\displaystyle \varphi(30) = \varphi(2 \times 3 \times 5) = \varphi(2)\varphi(3)\varphi(5) = 2^0(2 - 1)3^0(3 - 1)5^0(5 - 1) = \textcolor{black}{\bold{8}}\)