N divided by the largest squarefree divisor of n

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The n divided by the largest squarefree divisor of n (n divided by the squarefree kernel of n) function is

${\frac {n}{{\rm {rad}}(n)}}=n\prod _{{p\mid n} \atop {p~{\rm {prime}}}}{\frac {1}{p}},\,$

where $\scriptstyle {\rm {rad}}(n)\,$ is the squarefree kernel of $\scriptstyle n\,$.

This is a multiplicative arithmetic function, e.g.

${\frac {mn}{{\rm {rad}}(mn)}}={\frac {m}{{\rm {rad}}(m)}}\cdot {\frac {n}{{\rm {rad}}(n)}},\ (m,n)\,=\,1,\,$

where $\scriptstyle (m,\,n)\,$ is the gcd function.

Related formulae

The compositorial of $\scriptstyle n\,$ is given by:

${\frac {n!}{n\#}}={\frac {n!}{{\rm {rad}}(n!)}}.\,$

Sequences

A003557 $\scriptstyle n\,$ divided by largest squarefree divisor of $\scriptstyle n,\ n\,\geq \,1\,$.

{1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, ...}

See also

• Squarefree kernel
• Compositorial