Jordan's Totient Function - Maple Help

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NumberTheory

 JordanTotient
 Jordan's totient function

 Calling Sequence JordanTotient( k, n )

Parameters

 k - positive integer n - positive integer

Description

 • 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)).

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
 > $\mathrm{JordanTotient}\left(1,8\right)=\mathrm{Totient}\left(8\right)$
 ${4}{=}{4}$ (1)
 > $\mathrm{JordanTotient}\left(2,8\right)\ne \mathrm{Totient}\left(8\right)$
 ${48}{\ne }{4}$ (2)
 > $\mathrm{JordanTotient}\left(2,8\right)\mathrm{JordanTotient}\left(2,9\right)=\mathrm{JordanTotient}\left(2,8\cdot 9\right)$
 ${3456}{=}{3456}$ (3)
 > $\mathrm{seq}\left(\mathrm{JordanTotient}\left(k,6\right),k=1..10\right)$
 ${2}{,}{24}{,}{182}{,}{1200}{,}{7502}{,}{45864}{,}{277622}{,}{1672800}{,}{10057502}{,}{60406104}$ (4)

The following commands plot the values of JordanTotient[k](n) for n from $2$ to $1000$, and for k from $2$ to $5$.

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

The following command plots the values of JordanTotient[k](4) for k from $1$ to $100$ using a logarithmic scale on the vertical axis.

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

Compatibility

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