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# Squarefree kernel

The squarefree kernel of a positive integer ${\displaystyle \scriptstyle n\,}$ (or radical of n, rad(n)) is defined as the product of distinct prime factors of n (i.e. the largest squarefree number dividing n)

${\displaystyle {\text{sqf}}(n)\equiv {\text{rad}}(n):=\prod _{{p\mid n} \atop {p~{\text{prime}}}}p.\,}$

 257     ${\displaystyle \scriptstyle {\text{rad}}(n)\,}$
0

In the above graph, notice how the squarefree numbers (with asymptotic density ${\displaystyle \scriptstyle {\frac {1}{\zeta (2)}}\,=\,{\frac {6}{{\pi }^{2}}}\,\approx \,0.60792710185403\ldots \,}$) stand out.

A007947 The largest squarefree number dividing n (the squarefree kernel of ${\displaystyle \scriptstyle n\,}$, the radical of ${\displaystyle \scriptstyle n\,}$)

{1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, ...}

## Properties

${\displaystyle {\rm {rad}}(m,n)={\rm {rad}}(m)\cdot {\rm {rad}}(n),\ (m,n)\,=\,1.\,}$

### Trivial formulas (most trivial first)

• ${\displaystyle {\rm {rad}}({\rm {rad}}(n))={\rm {rad}}(n)\,}$      (Idempotence[1])

• ${\displaystyle {\rm {q}}(n)=[{\rm {rad}}(n)=n]\,}$      (where ${\displaystyle \scriptstyle [\cdot ]\,}$ is the Iverson bracket)

• ${\displaystyle \Omega ({\rm {rad}}(n))=\omega ({\rm {rad}}(n))=\omega (n)\,}$

• ${\displaystyle \mu ({\rm {rad}}(n))={(-1)}^{\omega ({\rm {rad}}(n))}={(-1)}^{\omega (n)}\,}$

• ${\displaystyle \mu (n)=\mu ({\rm {rad}}(n))\cdot {\rm {q}}(n)\,}$

• ${\displaystyle \sigma _{0}({\rm {rad}}(n))=\tau ({\rm {rad}}(n))={2}^{\omega (n)}\,}$

• ${\displaystyle \sigma _{1}({\rm {rad}}(n))=\psi ({\rm {rad}}(n))\,}$      (Cf. A048250)
where ${\displaystyle \psi (n)\,\equiv \,n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\frac {\sigma _{1}(p)}{p}}}={\frac {n}{{\rm {rad}}(n)}}\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\sigma _{1}(p)}}\,}$ is Dedekind psi function (Cf. A001615)

and thus ${\displaystyle \psi ({\rm {rad}}(n))={\frac {{\rm {rad}}(n)}{{\rm {rad}}({\rm {rad}}(n))}}\cdot {\prod _{p|{\rm {rad}}(n) \atop p~{\rm {prime}}}{\sigma _{1}(p)}}={\prod _{p|n \atop p~{\rm {prime}}}{\sigma _{1}(p)}}={\prod _{p|{\rm {rad}}(n) \atop p~{\rm {prime}}}{\sigma _{1}(p)}}=\sigma _{1}({\rm {rad}}(n))\,}$

and ${\displaystyle \psi (n)={\frac {n}{{\rm {rad}}(n)}}\cdot \psi ({\rm {rad}}(n))={\frac {n}{{\rm {rad}}(n)}}\cdot \sigma _{1}({\rm {rad}}(n))\,}$

• ${\displaystyle \psi ({\rm {rad}}(n))={\frac {\psi (n)\cdot {\rm {rad}}(n)}{n}}\,}$      (Cf. A048250)

• ${\displaystyle \psi (n)\cdot {\rm {rad}}(n)=n\cdot \psi ({\rm {rad}}(n))\,}$, or ${\displaystyle \ {\frac {\psi (n)}{n}}={\frac {\psi ({\rm {rad}}(n))}{{\rm {rad}}(n)}}\,}$

• ${\displaystyle \phi (n)\cdot {\rm {rad}}(n)=n\cdot \phi ({\rm {rad}}(n))\,}$, or ${\displaystyle \ {\frac {\phi (n)}{n}}={\frac {\phi ({\rm {rad}}(n))}{{\rm {rad}}(n)}}\,}$
where ${\displaystyle \phi ({\rm {rad}}(n))=\phi {\bigg (}\prod _{p|n \atop p~{\rm {prime}}}{p}{\bigg )}=\prod _{p|n \atop p~{\rm {prime}}}{\phi (p)}=\prod _{p|n \atop p~{\rm {prime}}}(p-1)={\rm {rad}}(n)\cdot \prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1-{\frac {1}{p}}{\bigg )}={\rm {rad}}(n)\cdot {\frac {\phi (n)}{n}}\,}$

• ${\displaystyle n\#={\rm {rad}}(n!)\,}$      (primorial of ${\displaystyle \scriptstyle n\,}$)

• ${\displaystyle {\frac {n!}{n\#}}={\frac {n!}{{\rm {rad}}(n!)}}\,}$      (compositorial of ${\displaystyle \scriptstyle n\,}$)

## n divided by the largest squarefree divisor of n

The n divided by the largest squarefree divisor of n function is

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

This is a multiplicative arithmetic function, e.g.

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

## Sequences

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

{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, ...}