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Squarefree numbers

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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