What is the equivalent of the integers symbol Z for n bit only integers?
For bounded sets, the usual convention is to use interval notation.

Specifically, $[a,b]$ means "real numbers between $a$ and $b$, inclusive", while $[a\mathinner{\ldotp \ldotp}b]$ means "integers between $a$ and $b$, inclusive". Changing any of the square brackets to curved means that endpoint is not included: $a \not\in (a,b]$, for example.

In this case, you'd want the interval $[0\mathinner{\ldotp \ldotp}2^n)$. Or, if you think the delimiters look mismatched and weird, you can use $[0\mathinner{\ldotp \ldotp}2^n-1]$.

P.S. LaTeX doesn't have a builtin for "two dots", so I used \mathinner{\ldotp \ldotp} from this answer.