Squarefree kernel

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The squarefree kernel of a positive integer $\scriptstyle n\,$ (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)

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

A007947: Product of distinct prime factors of $\scriptstyle n,\,1\,\leq \,n\,\leq \,257.\,$

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

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

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

{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

The squarefree kernel function (radical function) is a multiplicative arithmetic function, e.g.

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

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)

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

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

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

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

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

• $\sigma _{1}({\rm {rad}}(n))=\psi ({\rm {rad}}(n))\,$      (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 $\scriptstyle n\,$)

n divided by the largest squarefree divisor of n

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

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

This is a multiplicative arithmetic function, e.g.

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

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

• A036691 Compositorial numbers: product of first $\scriptstyle n\,$ composite numbers.
• A049614 Compositorial of $\scriptstyle n\,$ ($\scriptstyle n!\,$ divided by its squarefree kernel.)
• A007947 Largest squarefree number dividing $\scriptstyle n\,$ (the squarefree kernel, or radical, of $\scriptstyle n\,$).

• Factorial, primorial, compositorial
• N divided by the squarefree kernel of n
• Squarefree kernel of n!

• {{Squarefree kernel}} ({{radical}}, {{rad}}) arithmetic function template

• Distinct prime factors of n (prime factors of n (without multiplicity))
• Number of distinct prime factors of n (little omega(n))
• Sum of distinct prime factors of n (sodpf(n))
• Product of distinct prime factors of n (radical of n, rad(n)) (squarefree kernel of n)

• Prime factors of n (with multiplicity)
• Number of prime factors of n (with multiplicity) (big Omega(n))
• Sum of prime factors of n (with multiplicity) (sopf(n)) (integer log of n)
• Product of prime factors of n (with multiplicity) (n, positive integers)