The JordanTotient( k, n ) command computes Jordan's totient function, a generalization of the Euler totient. (See NumberTheory[Totient].) For positive integers $k$ and $n$, the Jordan totient JordanTotient( k, n ) is defined to be the number of $k$-tuples (a[1], a[2], ..., a[k]) of positive integers, each less than or equal to $n$, such that igcd( a[1], a[2], ..., a[k], n ) = 1.

For k = 1, we have JordanTotient( 1, n ) = Totient( n ).

For a fixed positive integer k, the Jordan totient is multiplicative in n; that is, if a and b are coprime positive integers, then JordanTotient( k, a*b ) = JordanTotient( k, a ) * JordanTotient( k, b ).

For a prime power n = p^a, we have JordanTotient( k, p^a ) = p^(k*a) - p^(k*(a-1)).

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

$\mathrm{JordanTotient}\left(1\,8\right)=\mathrm{Totient}\left(8\right)$

$4=4$

$\mathrm{JordanTotient}\left(2\,8\right)\ne \mathrm{Totient}\left(8\right)$

$48\ne 4$

$\mathrm{JordanTotient}\left(2\,8\right)\mathrm{JordanTotient}\left(2\,9\right)=\mathrm{JordanTotient}\left(2\,8\cdot 9\right)$

$3456=3456$

$\mathrm{seq}\left(\mathrm{JordanTotient}\left(k\,6\right)\,k=1..10\right)$

$2,24,182,1200,7502,45864,277622,1672800,10057502,60406104$

$P≔\left[\mathrm{seq}\right]\left(\mathrm{plots}:-\mathrm{pointplot}\left(\left[\mathrm{seq}\left(\left[n\,\mathrm{JordanTotient}\left(k\,n\right)\right]\,n=2..1000\right)\right]\,\mathrm{labels}=\left[''n''\,\mathrm{\phi }\left[k\right]\left(n\right)\right]\,\mathrm{color}=\mathrm{ColorTools}:-\mathrm{HueSpread}\left(''Blue''\,4\,\frac{1}{10}\right)\left[k-1\right]\,\mathrm{symbol}=\mathrm{circle}\right)\,k=2..5\right)\:$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{Array}\left(P\right)\right)$

$\mathrm{plots}:-\mathrm{logplot}\left(\left[\mathrm{seq}\left(\left[k\,\mathrm{JordanTotient}\left(k\,4\right)\right]\,k=1..100\right)\right]\,\mathrm{labels}=\left[''k''\,\mathrm{\phi }\left[k\right]\left(4\right)\right]\,\mathrm{color}=''Niagara\; BlueGreen''\right)$

The NumberTheory[JordanTotient] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.