Radical of an integer

In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n (each prime factor of n occurs exactly once as a factor of the product mentioned):

Radical numbers for the first few positive integers are

For example,

and therefore

The function ${\displaystyle \mathrm {rad} }$ is multiplicative (but not completely multiplicative).

The radical of any integer n is the largest square-free divisor of n and so also described as the square-free kernel of n. The definition is generalized to the largest t-free divisor of n, ${\displaystyle \mathrm {rad} _{t}}$, which are multiplicative functions which act on prime powers as

The cases t=3 and t=4 are tabulated in   and  .

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