By definition your distribution is discrete, because you can obtain all the values by counting.
Your confusion may stem from two sources. One is that often people assume that discrete also means finite. This is not true, e.g. the Poisson distribution is defined on the non-negative integers, which is an infinite countable set $[0,+\infty)$.
Another source could be usage of computer-generated random numbers, or pseudo random number generators. Since in computers even continuous variables, such as real numbers, are represented by countable sets (e.g. IEEE double precision floating point), the PRNGs generate discrete sequences to approximate continuous variables. So, in some sense, everything we do in computers is discrete.
By "computers" I mean digital computers. Analog computers are different.