Square-free integer
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| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Sociable number |
| Sublime number |
| Harmonic divisor number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The small square-free numbers are
- 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, ... (sequence A005117 in OEIS)
[edit] Equivalent characterizations of square-free numbers
The positive integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. Another way of stating the same is that for every prime multiple p of n, the prime p does not divide n / p. Yet another formulation: n is square-free if and only if in every factorization n=ab, the factors a and b are coprime.
The positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.
The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic. This follows from the classification of finitely generated abelian groups.
The integer n is square-free if and only if the factor ring Z / nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.
For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free.
The radical of an integer is always square-free.
[edit] Distribution of square-free numbers
If Q(x) denotes the number of square-free integers between 1 and x, then
(see pi and big O notation). The asymptotic/natural density of square-free numbers is therefore
where ζ is the Riemann zeta function.
Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show
[edit] Erdős Squarefree Conjecture
The central binomial coefficient
is never squarefree for n > 4. This was proven in 1996 by Olivier Ramaré and Andrew Granville.
