Q:

What is the standard notation to represent the set of primes?

I have seen $\mathbb Z_p$. Are there others, perhaps $\mathbb N_p$?
or the set of natural numbers where the totient of $ n $ equal $ n - 1 $ ?
$$ \{n \in \mathbb N \mid n\ge2,\phi (n) = n-1\}$$

A:

First: $\mathbb{Z}_p$ does not refer to the set of primes.  Depending on the context, it either refers to $\mathbb{Z}/p\mathbb{Z}$ or the $p$-adic integers.
The most common notation that I have seen for the set of primes is $\mathbb{P}$; however, it is not universally used, and so you should make sure to define it whenever you use it.

A:

$\mathbb{P}$ seems to be fairly common.

A:

It is surprisingly common to eschew any such notation and, instead, rely on a convention that $p$ (and $p'$, $p_n$ etc) always denotes a prime number.
In fact, you can see that convention used in a recent question right here.

