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# Radical of n, product of distinct prime factors of n

(Redirected from Squarefree kernel of n)

The radical (squarefree kernel, largest squarefree divisor) of an integer is the product of distinct prime factors of that integer. For example, the radical of 12 is 6, since 2 × 3 = 6. If an integer is squarefree, it is equal to its radical (see A007947).

Multiplicative with ${\displaystyle \scriptstyle a(p^{\alpha })\,=\,p,\,}$ thus with ${\displaystyle \scriptstyle n\,=\,\prod _{i=1}^{\omega (n)}p_{i}^{\alpha _{i}},\,}$ we have

${\displaystyle {\rm {rad}}(n)=\prod _{i=1}^{\omega (n)}p_{i}.\,}$

## n divided by radical of n

Multiplicative with ${\displaystyle \scriptstyle a(p^{\alpha })\,=\,p^{\alpha -1},\,}$ thus with ${\displaystyle \scriptstyle n\,=\,\prod _{i=1}^{\omega (n)}p_{i}^{\alpha _{i}},\,}$ we have

${\displaystyle {\frac {n}{{\rm {rad}}(n)}}=\prod _{i=1}^{\omega (n)}p_{i}^{\alpha _{i}-1}.\,}$

## Sequences

A007947 Largest squarefree number dividing ${\displaystyle \scriptstyle n\,}$, the squarefree kernel of ${\displaystyle \scriptstyle n\,}$, radical of ${\displaystyle \scriptstyle n\,}$: ${\displaystyle \scriptstyle {\rm {rad}}(n).\,}$

{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, ...}

A003557 ${\displaystyle \scriptstyle n\,}$ divided by radical (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, ...}