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

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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, ...}