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Calling Sequence
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JordanTotient( k, n )
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Parameters
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k
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-
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positive integer
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n
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-
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positive integer
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Description
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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.
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For k = 1, we have JordanTotient( 1, n ) = Totient( n ).
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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 ).
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For a prime power n = p^a, we have JordanTotient( k, p^a ) = p^(k*a) - p^(k*(a-1)).
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Examples
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The following commands plot the values of JordanTotient[k](n) for n from to , and for k from to .
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The following command plots the values of JordanTotient[k](4) for k from to using a logarithmic scale on the vertical axis.
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Compatibility
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The NumberTheory[JordanTotient] command was introduced in Maple 2020.
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