The Totient function computes Euler's totient function.

Given a positive integer n, Totient(n) returns the number of positive integers coprime to n and not greater than n.

phi and varphi are aliases of Totient.

You can enter the commands phi and varphi using either the 1-D or 2-D calling sequence. For example, phi(8) is equivalent to $\mathrm{\phi }\left(8\right)$, and varphi(8) is equivalent to $\mathrm{\varphi }\left(8\right)$.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\left[\mathrm{Totient}\left(1\right)\,\mathrm{\phi }\left(2\right)\,\mathrm{\varphi }\left(3\right)\right]$

$\left[1\,1\,2\right]$

$\left[\mathrm{Totient}\left(44\right)\mathrm{Totient}\left(79\right)\,\mathrm{Totient}\left(44\cdot 79\right)\right]$

$\left[1560\,1560\right]$

$\mathrm{andmap}\left(i\mapsto \mathrm{igcd}\left(i\left[1\right]\,i\left[2\right]\right)\ne 1\mathbf{xor}\mathrm{\phi }\left(i\left[1\right]\cdot i\left[2\right]\right)=\mathrm{\phi }\left(i\left[1\right]\right)\cdot \mathrm{\phi }\left(i\left[2\right]\right)\,\left[\mathrm{seq}\left(\mathrm{seq}\left(\left[i\,j\right]\,j=1..100\right)\,i=1..100\right)\right]\right)$

$\mathrm{true}$

$\left[\mathrm{Totient}\left(59\right)\,\mathrm{Totient}\left(101\right)\right]$

$\left[58\,100\right]$

$\mathrm{andmap}\left(i\mapsto \mathrm{ithprime}\left(i\right)-1=\mathrm{\varphi }\left(\mathrm{ithprime}\left(i\right)\right)\,\left[\mathrm{seq}\left(1..100\right)\right]\right)$

$\mathrm{true}$

$\mathrm{plots}:-\mathrm{pointplot}\left(\left[\mathrm{seq}\left(\left[n\,\mathrm{Totient}\left(n\right)\right]\,n=2..1000\right)\right]\,\mathrm{labels}=\left[''n''\,\phi \left(n\right)\right]\,\mathrm{color}=''Niagara\; BlueGreen''\,\mathrm{symbol}=\mathrm{circle}\right)$

The NumberTheory[Totient] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.