The product of the prime factors of a given 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 this product:
formula_1
The radical plays a central role in the statement of the abc conjecture.
Examples.
Radical numbers for the first few positive integers are
 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, ... (sequence in the OEIS).
For example,
formula_2
and therefore
formula_3
Properties.
The function formula_4 is multiplicative (but not completely multiplicative).
The radical of any integer formula_5 is the largest square-free divisor of formula_5 and so also described as the square-free kernel of formula_5. There is no known polynomial-time algorithm for computing the square-free part of an integer.
The definition is generalized to the largest formula_8-free divisor of formula_5, formula_10, which are multiplicative functions which act on prime powers as
formula_11
The cases formula_12 and formula_13 are tabulated in OEIS:  and OEIS: .
The notion of the radical occurs in the abc conjecture, which states that, for any formula_14, there exists a finite formula_15 such that, for all triples of coprime positive integers formula_16, formula_17, and formula_18 satisfying formula_19,
formula_20
For any integer formula_5, the nilpotent elements of the finite ring formula_22 are all of the multiples of formula_23.
The Dirichlet series is
formula_24
References.
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