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# User:Enrique Pérez Herrero/SquarefreeKernel

## SQUAREFREE KERNEL

### Trivial Formulas (most trivial first):

• ${\rm {rad}}({\rm {rad}}(n))={\rm {rad}}(n)\,$ ,   (idempotence)

• ${\rm {q}}({\rm {rad}}(n))=1\,$ , where ${\rm {q}}(n)\,$ is the characteristic function of squarefree numbers or quadratfrei function

• ${\rm {q}}(n)=\delta _{n}^{{\rm {rad}}(n)}\,$ , where $\delta _{i}^{j}\,$ is the Kronecker delta

• $\Omega ({\rm {rad}}(n))=\omega ({\rm {rad}}(n))=\omega (n)\,$ • $\mu ({\rm {rad}}(n))={(-1)}^{\omega ({\rm {rad}}(n))}={(-1)}^{\omega (n)}\,$ • $\mu (n)=\mu ({\rm {rad}}(n))\cdot {\rm {q}}(n)\,$ • $\sigma _{0}({\rm {rad}}(n))=\tau ({\rm {rad}}(n))={2}^{\omega (n)}\,$ • $\sigma _{1}({\rm {rad}}(n))=\psi ({\rm {rad}}(n))\,$ ,   (Cf. A048250)
where $\psi (n)\,\equiv \,n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\frac {\sigma _{1}(p)}{p}}}={\frac {n}{{\rm {rad}}(n)}}\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\sigma _{1}(p)}}\,$ is Dedekind psi function (Cf. A001615)

and thus $\psi ({\rm {rad}}(n))={\frac {{\rm {rad}}(n)}{{\rm {rad}}({\rm {rad}}(n))}}\cdot {\prod _{p|{\rm {rad}}(n) \atop p~{\rm {prime}}}{\sigma _{1}(p)}}={\prod _{p|n \atop p~{\rm {prime}}}{\sigma _{1}(p)}}={\prod _{p|{\rm {rad}}(n) \atop p~{\rm {prime}}}{\sigma _{1}(p)}}=\sigma _{1}({\rm {rad}}(n))\,$ and $\psi (n)={\frac {n}{{\rm {rad}}(n)}}\cdot \psi ({\rm {rad}}(n))={\frac {n}{{\rm {rad}}(n)}}\cdot \sigma _{1}({\rm {rad}}(n))\,$ • $\psi ({\rm {rad}}(n))={\frac {\psi (n)\cdot {\rm {rad}}(n)}{n}}\,$ , (Cf. A048250)

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

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

### Mathematica Code:

rad[n_]:=Times@@(FactorInteger[n][[All,1]]);