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# Squarefree numbers

Squarefree numbers are numbers not divisible by a square greater than 1. Alternately, they are numbers with all exponents in its prime factorization less than 2. Note that although 1 is a square, it is also squarefree. The squarefree numbers are sequence A005117, and the first few squarefree 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, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, ...

## Characteristic function

The characteristic function of squarefree numbers is given by

where is the Möbius function. When is squarefree and otherwise The first few terms are

- 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, ... (A008966)

Alternately,

- ,

being the sign function, or

- , where is the Kronecker delta and is the radical or squarefree kernel of .

## Squarefree counting function

The summatory quadratfrei function is defined as

The asymptotic density of squarefree numbers corresponds to the probability that 2 randomly chosen integers are coprime

where is the Riemann zeta function.

## See also

- A007947 Largest squarefree number dividing n (the squarefree kernel of n).
- A007913 Squarefree part of n: a(n) = smallest positive number m such that n/m is a square.
- A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.