A positive integer is squarefull when every prime dividing it also divides it to at least the second power. Thus is not squarefull as soon as some prime satisfies but .
Every positive integer has a unique factorization with powerful, squarefree, and : assign each prime power of exponent at least two to , and each prime of exponent one to . In general differs from the squarefree part, which instead retains every prime of odd exponent.
A powerful number that is not a perfect power. If is its prime factorization, the conditions are for every and .
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A **powerful number** is a positive integer \( n \) such that for every prime \( p \) that divides \( n \), \( p^2 \) also divides \( n \). In other words, if a prime number appears in the factorization of a powerful number, it must appear with an exponent of at least 2.