Neliövapaa

Neliövapaa is a Finnish term that translates to "square-free" in English, primarily used in number theory. A positive integer is considered neliövapaa if it is not divisible by any perfect square greater than 1. This means that in its prime factorization, no prime appears with an exponent greater than 1. For example, the numbers 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, and 15 are neliövapaa. In contrast, 4 is not neliövapaa because it is divisible by 2^2 = 4. Similarly, 8 is not neliövapaa as it is divisible by 2^2 = 4, and 12 is not neliövapaa because it is divisible by 2^2 = 4.

The concept of neliövapaa numbers is fundamental in various areas of mathematics, including abstract algebra and combinatorial number theory. The density of neliövapaa numbers among the positive integers is approximately 6/pi^2, which is about 60.8%. A number n is neliövapaa if and only if its Möbius function, mu(n), is not zero. The Möbius function is defined as mu(n) = 1 if n is a square-free positive integer with an even number of prime factors, mu(n) = -1 if n is a square-free positive integer with an odd number of prime factors, and mu(n) = 0 if n has a squared prime factor.