Characteristic functions

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A characteristic function (or indicator function)

$\chi _{A}(x)\equiv \chi _{p(A)}(x)\equiv \mathbf {I} _{A}(x)\equiv \mathbf {I} _{p(A)}(x)\,$

returns 1 if $\scriptstyle x\,$ has the defining property $\scriptstyle p(A)\,$ of a given set $\scriptstyle A\,$ or 0 otherwise.

For example, the characteristic function of the primes $\scriptstyle \chi _{\rm {prime}}(n)\,=\,[\Omega (n)\,=\,1]\,=\,\pi (n)-\pi (n-1)\,$ yields 1 only when $\scriptstyle n\,$ is prime (see A010051). Another example is the characteristic function of squarefree numbers $\scriptstyle q(n)\,=\,|\mu (n)|\,$ (see A008966).

See Index to OEIS: Section Ch for a thorough listing of characteristic function sequences in the OEIS.