A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$
1. **Understanding the Problem:**
   We need to find the number of positive integers \( n \) such that \( n \geq 4 \) and there exists a complex number \( z \) with \( |z| = 1 \) satisfying the equation:
   \[
   1 + z + z^2 + z^{n-1} + z^n = 0.
   \]

2. **Analyzing the Equation:**
   Given \( |z| = 1 \), we know \( z \) lies on the unit circle in the complex plane. This implies \( z \cdot \overline{z} = 1 \).

3. **Taking the Conjugate:**
   Taking the complex conjugate of the given equation:
   \[
   \overline{1 + z + z^2 + z^{n-1} + z^n} = 0.
   \]
   Since the conjugate of a sum is the sum of the conjugates, we have:
   \[
   1 + \overline{z} + \overline{z}^2 + \overline{z}^{n-1} + \overline{z}^n = 0.
   \]
   Using \( \overline{z} = \frac{1}{z} \), we get:
   \[
   1 + \frac{1}{z} + \frac{1}{z^2} + \frac{1}{z^{n-1}} + \frac{1}{z^n} = 0.
   \]
   Multiplying through by \( z^n \) to clear the denominators:
   \[
   z^n + z^{n-1} + z^{n-2} + z + 1 = 0.
   \]

4. **Subtracting the Original Equation:**
   Subtracting the original equation from this new equation:
   \[
   (z^n + z^{n-1} + z^{n-2} + z + 1) - (1 + z + z^2 + z^{n-1} + z^n) = 0,
   \]
   simplifies to:
   \[
   z^{n-2} - z^2 = 0.
   \]
   This implies:
   \[
   z^{n-2} = z^2 \implies z^{n-4} = 1.
   \]

5. **Roots of Unity:**
   Since \( z^{n-4} = 1 \), \( z \) must be a root of unity. Let \( k \) be the smallest positive integer such that \( z^k = 1 \). Then \( k \) divides \( n-4 \).

6. **Form of \( n \):**
   For \( z \) to be a 5th root of unity, \( n-4 \) must be a multiple of 5. Therefore, \( n \) can be written as:
   \[
   n = 5m + 4,
   \]
   where \( m \) is a non-negative integer.

7. **Counting the Values:**
   We need to count the number of integers \( n \) of the form \( 5m + 4 \) that are less than or equal to 2022. Solving for \( m \):
   \[
   5m + 4 \leq 2022 \implies 5m \leq 2018 \implies m \leq \frac{2018}{5} \implies m \leq 403.6.
   \]
   Since \( m \) must be an integer, the largest possible value for \( m \) is 403. Therefore, \( m \) ranges from 0 to 403, inclusive.

8. **Number of Values:**
   The number of integers \( m \) in this range is:
   \[
   403 - 0 + 1 = 404.
   \]

The final answer is \(\boxed{404}\).