Definitions Let $A\subset\omega$. $A$ is $n$-r.e. if there is a computable function $f(a,s)$ such that for all $a\in\omega$,
$f(a,0)=0$, $|\{s:f(a,s)\neq f(a,s+1)\}|\leq n$, $\lim_{s\to\infty}f(a,s)=\chi_A(a)$. Note that $1$-r.e. sets are just r.e. sets; $2$-r.