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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

 > with(NumberTheory):
 > JordanTotient( 1, 8 ) = Totient( 8 );
 ${4}{=}{4}$ (1)
 > JordanTotient( 2, 8 ) <> Totient( 8 );
 ${48}{\ne }{4}$ (2)
 > JordanTotient( 2, 8 ) * JordanTotient( 2, 9 ) = JordanTotient( 2, 8*9 );
 ${3456}{=}{3456}$ (3)
 > seq( JordanTotient( k, 6 ), k = 1 .. 10 );
 ${2}{,}{24}{,}{182}{,}{1200}{,}{7502}{,}{45864}{,}{277622}{,}{1672800}{,}{10057502}{,}{60406104}$ (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 := [seq]( plots:-pointplot([seq([n, JordanTotient( k, n )], n = 2..1000)], labels = ["n",φ[k](n)], color = ColorTools:-HueSpread( "Blue", 4, 1/10 )[ k - 1 ],  symbol = circle), k = 2 .. 5 ):
 > plots:-display( Array( P ) );

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

 > plots:-logplot([seq([k, JordanTotient( k, 4 )], k = 1..100)], labels = ["k",φ[k](4)], color = "Niagara BlueGreen");
 >

Compatibility

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