Squarefreeness

Squarefree numbers are positive integers that are not divisible by the square of any prime. More precisely, a positive integer n is squarefree if in its prime factorization n = p1^e1 p2^e2 ... pr^er every ei is either 0 or 1; equivalently, no prime p satisfies p^2 | n. By convention, 1 is squarefree, while 0 is typically not considered squarefree.

Examples and non-examples illustrate the idea. Squarefree numbers include 1, 2, 3, 5, 6, 10, 14, 15, 21, and 30. Non-examples include 4 (2^2 divides 4), 8 (2^3), 9 (3^2), 12 (2^2 divides 12), 18 (3^2 divides 18), and 20 (2^2 divides 20).

Squarefreeness has several important properties. The Möbius function μ(n) encodes squarefreeness: μ(n) = 0 if n has a squared prime factor, and μ(n) = (-1)^k if n is a product of k distinct primes. The set of squarefree numbers has natural density 6/π^2, so the number of squarefree integers up to x is asymptotically (6/π^2) x. The radical function rad(n) equals the product of distinct primes dividing n; for a squarefree n, rad(n) = n.

Testing and computation follow from these characterizations. To check squarefreeness, factor n and verify that all exponents are at most 1, or test divisibility by p^2 for primes p ≤ √n. Efficient sieving methods can enumerate or count squarefree numbers up to a bound.

Applications and related concepts include Möbius inversion, discriminants in algebra, and various counting problems in analytic number theory. Squarefreeness also plays a role in the study of arithmetic functions and in probabilistic models of integers.