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$n$-free number (Definition)

The concept of a squarefree number can be generalized. Let $ n \in \mathbb{Z}$ with $ n>1$. Then $ m \in \mathbb{Z}$ is $ n$-free if, for any prime $ p$, $ p^n$ does not divide $ m$.

Let $ S$ denote the set of all squarefree natural numbers. Note that, for any $ n$ and any positive $ n$-free integer $ m$, there exists a unique $ (a_1, \dots , a_{n-1}) \in S^{n-1}$ with $ \gcd(a_i,a_j)=1$ for $ i \neq j$ such that $ \displaystyle m=\prod_{j=1}^{n-1} {a_j}^j$.



"$n$-free number" is owned by Wkbj79.
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See Also: square-free number, $n$-full number

Also defines:  cubefree, cubefree number, cube free, cube free number, cube-free, cube-free number
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Cross-references: integer, positive, natural numbers, divide, prime, number, squarefree
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This is version 3 of $n$-free number, born on 2006-06-26, modified 2006-08-19.
Object id is 8089, canonical name is NFreeNumber.
Accessed 2222 times total.

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AMS MSC: 11A51 (Number theory :: Elementary number theory :: Factorization; primality)
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