Square-free

In mathematics, a square-free integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The small square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, ... (sequence A005117 in OEIS)

Equivalent characterizations of square-free numbers

The integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. Another way of stating the same is that for every prime divisor p of n, the prime p does not divide n / p. Yet another formulation: n is square-free if and only if in every factorization n=ab, the factors a and b are coprime.

The positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.

The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic. This follows from the classification of finitely generated abelian groups.

The integer n is square-free iff the factor ring Z / nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.

For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free.

Given the positive integer n, define the radical of the integer n by

equal to the product of the prime numbers p dividing n. Then the square-free n are exactly the solutions of n = rad(n).

Distribution of square-free numbers

If Q(x) denotes the number of square-free integers between 1 and x, then

[itex]Q(x) = \frac{6x}{\pi^2} + O(\sqrt{x})[itex]

(see pi and big O notation). The asymptotic/natural density of square-free numbers is therefore

[itex]\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2} = \frac{1}{\zeta(2)}[itex]

where ζ is the Riemann zeta function.

Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show

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