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 $\scriptstyle {\rm {q}}(n)\,$ is the characteristic function of squarefree numbers or quadratfrei function

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

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

Mathematica Code:

rad[n_]:=Times@@(FactorInteger[n][[All,1]]);
A007947[n_]:=rad[n];
SetAttributes[{rad,A007947},Listable];
rad::usage="rad[n]: Largest squarefree number dividing n (the squarefree kernel of n).";
A007947::usage="A007947[n]: Largest squarefree number dividing n (the squarefree kernel of n).";

Links:

• Squarefree kernel
• Squarefree numbers