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# Characteristic function of squarefree numbers

The quadratfrei function is the characteristic function of squarefree numbers.

The quadratfrei function is given by

where (mu(n)) is the Möbius function. (When is a squarefree number we have , otherwise when is a squareful number we have .)

The quadratfrei function is also given by

where is the number of prime factors of n (with multiplicity), is the number of distinct prime factors of n and where is the Iverson bracket. Also

where is the radical or squarefree kernel of .

## Contents

## Related arithmetic functions

### Characteristic function of nonsquarefree numbers

The characteristic function of nonsquarefree numbers, i.e. the complement of the quadratfrei function is given by

where is the Iverson bracket.

### Summatory quadratfrei function

The summatory quadratfrei function is defined as

where is the quadratfrei function (characteristic function of squarefree numbers) and is the Moebius function.

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

where is the ^{th} prime number, and is the Riemann zeta function.

The asymptotic density of squarefree numbers with an odd number of prime factors is equal to the the asymptotic density of squarefree numbers with an even number of prime factors, i.e.

where is the Iverson bracket.

The graph of the Mertens function (the Mertens function being the summatory Moebius function) seems to indicate an average negative bias for the Mertens function, which would mean that there is a bias (eerily similar to the Chebyshev bias) in favor of the squarefree numbers with an odd number of prime factors over the squarefree numbers with an even number of prime factors. This existence or not of such a bias, if small enough, would have no effect on the asymptotic behavior.

## Sequences

(Cf. A008966) gives the sequence

- {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, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...}

(Cf. A107078) gives the sequence

- {0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, ...}

## See also

- Omega(n), number of prime factors of n (with multiplicity).
- A001222 Number of prime divisors of n (counted with multiplicity).
- omega(n), number of distinct prime factors of n.
- A001221 Number of distinct primes dividing n (also called omega(n)).
- A046660 Excess of n = number of primes divisors (with multiplicity) - number of prime divisors (without multiplicity).
- A107078 Indicates that n has non-unitary prime divisors. Characteristic function of nonsquarefree numbers.
- A212793 Characteristic function of cubefree numbers.