In a sequence of natural numbers $ a_1,a_2,...,a_n$ every number $ a_k$ represents sum of the multiples of the $ k$ from sequence. Find all possible values for $ n$.
1. Let's start by understanding the problem statement. We are given a sequence of natural numbers \( a_1, a_2, \ldots, a_n \) where each \( a_k \) represents the sum of the multiples of \( k \) from the sequence. This means:
   \[
   a_k = \sum_{i \in S_k} a_i
   \]
   where \( S_k = \{ j \neq k : k \mid a_j \} \).

2. To find all possible values for \( n \), we need to analyze the conditions under which the sequence can exist. Let's consider the simplest case where \( n = 2 \).

3. For \( n = 2 \), we have the sequence \( (a_1, a_2) \). According to the problem, \( a_1 \) should be the sum of all multiples of 1 in the sequence, and \( a_2 \) should be the sum of all multiples of 2 in the sequence. Since every number is a multiple of 1, we have:
   \[
   a_1 = a_1 + a_2
   \]
   This simplifies to:
   \[
   a_1 - a_1 = a_2 \implies 0 = a_2
   \]
   Therefore, \( a_2 = 0 \).

4. Next, we need to check if \( a_2 \) satisfies the condition. Since \( a_2 \) is the sum of all multiples of 2 in the sequence, and the only multiple of 2 in the sequence is \( a_2 \) itself, we have:
   \[
   a_2 = a_2
   \]
   This is trivially true.

5. Therefore, for \( n = 2 \), the sequence \( (a_1, a_2) = (a_1, 0) \) works, where \( a_1 \) can be any natural number.

6. Now, let's consider \( n = 3 \). We have the sequence \( (a_1, a_2, a_3) \). According to the problem, we have:
   \[
   a_1 = a_1 + a_2 + a_3
   \]
   This simplifies to:
   \[
   a_1 - a_1 = a_2 + a_3 \implies 0 = a_2 + a_3
   \]
   Therefore, \( a_2 + a_3 = 0 \).

7. For \( a_2 \), we have:
   \[
   a_2 = \sum_{i \in S_2} a_i
   \]
   The multiples of 2 in the sequence are \( a_2 \) and \( a_3 \) (if \( a_3 \) is even). Since \( a_2 \) is a multiple of 2, we have:
   \[
   a_2 = a_2 + a_3
   \]
   This simplifies to:
   \[
   a_2 - a_2 = a_3 \implies 0 = a_3
   \]
   Therefore, \( a_3 = 0 \).

8. For \( a_3 \), we have:
   \[
   a_3 = \sum_{i \in S_3} a_i
   \]
   The multiples of 3 in the sequence are \( a_3 \) (if \( a_3 \) is a multiple of 3). Since \( a_3 = 0 \), this condition is trivially satisfied.

9. Therefore, for \( n = 3 \), the sequence \( (a_1, a_2, a_3) = (a_1, 0, 0) \) works, where \( a_1 \) can be any natural number.

10. By induction, we can generalize this result for any \( n \). For any \( n \), the sequence \( (a_1, a_2, \ldots, a_n) = (a_1, 0, 0, \ldots, 0) \) works, where \( a_1 \) can be any natural number.

The final answer is \( \boxed{ n } \) can be any natural number.