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JordanTotient

  

Jordan's totient function

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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

withNumberTheory:

JordanTotient1,8=Totient8

4=4

(1)

JordanTotient2,8Totient8

484

(2)

JordanTotient2,8JordanTotient2,9=JordanTotient2,89

3456=3456

(3)

seqJordanTotientk,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.

Pseqplots:-pointplotseqn,JordanTotientk,n,n=2..1000,labels=n,φkn,color=ColorTools:-HueSpreadBlue,4,110k1,symbol=circle,k=2..5:

plots:-displayArrayP

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:-logplotseqk,JordanTotientk,4,k=1..100,labels=k,φk4,color=Niagara BlueGreen

Compatibility

• 

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

• 

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

See Also

NumberTheory

NumberTheory[Totient]