The Set of Natural Numbers

$\N$

$\N = \left\{{0, 1, 2, 3, \ldots}\right\}$.

Its $\LaTeX$ code is \mathbb N or \N.

The Set of Non-Zero Natural Numbers

$\N^*$

$\N^* = \left\{{1, 2, 3, \ldots}\right\}$.

Its $\LaTeX$ code is \mathbb N^* or \N^*.

$\N_n$, $\N^*_n$

The set $\N_n$ is the set of all natural numbers which are less than $n$:

$\N_n = \left\{{x \in \N: x < n}\right\} = \left\{{0, 1, 2, \ldots, n-1}\right\}$.

Its $\LaTeX$ code is \mathbb N_n or \N_n.

Similarly, the set $\N^*_n$ is the set of all non-zero natural numbers which are less or equal to $n$:

$\N^*_n = \left\{{x \in \N^*: x \le n}\right\} = \left\{{1, 2, \ldots, n}\right\}$.

Its $\LaTeX$ code is \mathbb N^*_n or \N^*_n.

Older literature tends to use $\N$ to mean $\left\{{1, 2, 3, \ldots}\right\}$.

Consequently, the set $\left\{{0, 1, 2, 3, \ldots}\right\}$ needs another symbol to denote it. The usual technique is to use $\tilde {\N}$.

Its $\LaTeX$ code is \tilde {\mathbb N} or \tilde {\N}.