Foleyszámokat

Foleyszámokat is a Hungarian term that translates to "folly numbers" or "fool's numbers." It refers to a set of numbers that have a peculiar property related to their prime factorization. Specifically, a Foleyszám is a number n such that if you take the sum of the exponents of the prime factors of n, the result is equal to the sum of the prime factors themselves.

For example, consider the number 12. Its prime factorization is $2^2 \times 3^1$. The sum of the exponents is $2 + 1 = 3$. The sum of the prime factors is $2 + 3 = 5$. Since 3 is not equal to 5, 12 is not a Foleyszám.

Now consider the number 30. Its prime factorization is $2^1 \times 3^1 \times 5^1$. The sum of the exponents is $1 + 1 + 1 = 3$. The sum of the prime factors is $2 + 3 + 5 = 10$. Again, these are not equal.

A known example of a Foleyszám is 1. The prime factorization of 1 is considered to be an empty product, meaning there are no prime factors. The sum of the exponents is 0, and the sum of the prime factors is also 0. Therefore, 1 is a Foleyszám.

Another example is the number 4. Its prime factorization is $2^2$. The sum of the exponents is 2. The sum of the prime factors is 2. Thus, 4 is a Foleyszám.

The study of Foleyszámokat is an area of number theory, though it is not as widely known or researched as other topics. The exact origin of the term "Foleyszámokat" is unclear, but it suggests a connection to curious or perhaps seemingly trivial mathematical properties. Discovering and classifying these numbers involves careful examination of prime factorizations.