Radical of an integer

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In mathematics, the radical of a positive integer n is defined as the product of the prime numbers dividing n:

\displaystyle\mbox{rad}(n)=\prod_{p|n}p.\,

For example,

504=2^3\cdot3^2\cdot7 \mbox{ and } \mbox{rad}(504)=2\cdot3\cdot7=42.\,

The radical of any integer n is the largest square-free divisor of n. For this reason the radical of n is also called the square-free part of n or the square-free kernel of n.

Radical numbers for the first few positive integers are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in OEIS).

The function rad is multiplicative.

One of the most striking applications of the notion of radical occurs in the abc conjecture.

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