The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$,  $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.
To solve the problem, we need to determine for which values of \( n \) the number \( a_n \) is not divisible by 3. The number \( a_n \) is formed by concatenating all natural numbers from 1 to \( n \).

1. **Understanding the structure of \( a_n \)**:
   - \( a_n \) is the concatenation of all natural numbers from 1 to \( n \).
   - For example, \( a_1 = 1 \), \( a_2 = 12 \), \( a_3 = 123 \), and so on.

2. **Sum of digits modulo 3**:
   - A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
   - Therefore, we need to find the sum of the digits of \( a_n \) and check its divisibility by 3.

3. **Sum of digits of numbers from 1 to \( n \)**:
   - Let's denote the sum of the digits of a number \( k \) as \( S(k) \).
   - The sum of the digits of \( a_n \) is the sum of the digits of all numbers from 1 to \( n \), i.e., \( S(1) + S(2) + \cdots + S(n) \).

4. **Using properties of modulo arithmetic**:
   - We need to find when the sum \( S(1) + S(2) + \cdots + S(n) \) is not divisible by 3.
   - Notice that the sum of the first \( n \) natural numbers is given by \( \frac{n(n+1)}{2} \).

5. **Sum of digits modulo 3**:
   - The sum of the digits of \( a_n \) modulo 3 can be related to the sum of the first \( n \) natural numbers modulo 3.
   - We need to check when \( \frac{n(n+1)}{2} \equiv 0 \pmod{3} \).

6. **Analyzing \( \frac{n(n+1)}{2} \pmod{3} \)**:
   - We need to find \( n \) such that \( \frac{n(n+1)}{2} \not\equiv 0 \pmod{3} \).
   - Consider the possible values of \( n \mod 3 \):
     - If \( n \equiv 0 \pmod{3} \), then \( n(n+1) \equiv 0 \cdot 1 \equiv 0 \pmod{3} \).
     - If \( n \equiv 1 \pmod{3} \), then \( n(n+1) \equiv 1 \cdot 2 \equiv 2 \pmod{3} \).
     - If \( n \equiv 2 \pmod{3} \), then \( n(n+1) \equiv 2 \cdot 3 \equiv 0 \pmod{3} \).

7. **Conclusion**:
   - From the above analysis, \( \frac{n(n+1)}{2} \equiv 0 \pmod{3} \) when \( n \equiv 0 \pmod{3} \) or \( n \equiv 2 \pmod{3} \).
   - Therefore, \( a_n \) is not divisible by 3 when \( n \equiv 1 \pmod{3} \).

The final answer is \( \boxed{ n \equiv 1 \pmod{3} } \).