In mathematics, a square-free 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, ... (sequence A005117 in OEIS) Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ... Look up one in Wiktionary, the free dictionary 1 (one) is a number, numeral, and glyph. ... For the Austin Powers character, see Number 2(Austin Powers 2 (two) is a number, numeral, and glyph. ... 3 (three) is a number, numeral, and glyph. ... 5 (five) is a number, numeral, and glyph. ... 6 (six) is the natural number following 5 and preceding 7. ... 7 (seven) is the natural number following 6 and preceding 8. ... 10 (ten) is the natural number following 9 and preceding 11. ... 11 (number) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... 13 (Thirteen) is the natural number following 12 and preceding 14. ... 14 (fourteen) is the natural number following 13 and preceding 15. ... 15 (fifteen) is the natural number following 14 and preceding 16. ... 17 (seventeen) is the natural number following 16 and preceding 18. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... 21 (twenty-one) is the natural number following 20 and preceding 22. ... 22 (twenty-two) is the natural number following 21 and preceding 23. ... 23 (twenty-three) is the natural number following 22 and preceding 24. ... 26 (twenty-six) is the natural number following 25 and preceding 27. ... 29 (twenty-nine) is the natural number following 28 and preceding 30. ... 30 (thirty) is the natural number following 29 and preceding 31. ... 31 is the natural number following 30 and preceding 32. ... 33 is the natural number following 32 and preceding 34. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

Equivalent characterizations of square-free numbers

The 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 divisor 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. In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Coprime - Wikipedia /**/ @import /skins-1. ...

The positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

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. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

The integer n is square-free iff 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. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. ...

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. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ... In formal logic, mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set-theoretic operations intersection, union and complement. ...

Given the positive integer n, define the radical of the integer n by

m = rad(n),

equal to the product of the prime numbers p dividing n. Then the square-free n are exactly the solutions of n = rad(n).

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 The minuscule, or lower-case, pi The mathematical constant π represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ... In mathematics, a sequence a1, a2, ... , an, with the aj positive integers and aj < aj+1 for all j, has natural density α, where 0 ≤ α ≤ 1, if the proportion of natural numbers included as some aj is asymptotic to α. More formally, if we define the counting function A(x) as the...

where ζ is the Riemann zeta function. In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...

Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show