quadratfrei

Quadratfrei is a term used in number theory to describe an integer that is not divisible by any perfect square other than 1. This means that in the prime factorization of a quadratfrei number, no prime factor appears with an exponent greater than 1. For example, 10 is quadratfrei because its prime factorization is 2 * 5, with each prime appearing only once. However, 12 is not quadratfrei because it is divisible by 4, which is 2 squared (2^2 * 3). Similarly, 18 is not quadratfrei as it is divisible by 9, which is 3 squared (2 * 3^2).

The concept of quadratfrei numbers is important in various areas of mathematics, including abstract algebra and combinatorics. For instance, in ring theory, a square-free integer plays a role in the study of unique factorization domains. In combinatorics, quadratfrei numbers appear in the context of counting problems and generating functions.

A number that is not quadratfrei is said to be "squareful". Every integer greater than 1 can be uniquely expressed as a product of a quadratfrei number and a squareful number. This decomposition is useful for simplifying certain number-theoretic problems. The density of quadratfrei numbers among the positive integers is approximately 6/pi^2, which is about 60.8%.