Jordan's Totient Function - Maple Help
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NumberTheory

  

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  and , the Jordan totient JordanTotient( k, n ) is defined to be the number of -tuples (a[1], a[2], ..., a[k]) of positive integers, each less than or equal to , 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

(1)

(2)

(3)

(4)

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

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

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]

 


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