NumberTheory
JordanTotient
Jordan's totient function
Calling Sequence
Parameters
Description
Examples
Compatibility
JordanTotient( k, n )
k
-
positive integer
n
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)).
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.
The NumberTheory[JordanTotient] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
See Also
NumberTheory[Totient]
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