Find all numbers that can be expressed in exactly $2010$ different ways as the sum of  powers of two with non-negative exponents, each  power appearing as a summand at most three times. A sum can also be made from just one summand.
To solve this problem, we need to find all numbers that can be expressed in exactly 2010 different ways as the sum of powers of two with non-negative exponents, where each power can appear as a summand at most three times.

1. **Understanding the Problem:**
   - We need to express a number \( n \) as a sum of powers of two.
   - Each power of two can appear at most three times in the sum.
   - We need to count the number of distinct ways to form such sums and find \( n \) for which this count is exactly 2010.

2. **Generating Functions:**
   - We can use generating functions to count the number of ways to form sums with the given constraints.
   - The generating function for each power of two \( 2^k \) appearing at most three times is:
     \[
     1 + x^{2^k} + x^{2 \cdot 2^k} + x^{3 \cdot 2^k}
     \]
   - The overall generating function is the product of these individual generating functions:
     \[
     G(x) = \prod_{k=0}^{\infty} (1 + x^{2^k} + x^{2 \cdot 2^k} + x^{3 \cdot 2^k})
     \]

3. **Counting the Number of Ways:**
   - The coefficient of \( x^n \) in the expansion of \( G(x) \) gives the number of ways to express \( n \) as a sum of powers of two with each power appearing at most three times.
   - We need to find \( n \) such that this coefficient is exactly 2010.

4. **Simplifying the Problem:**
   - Instead of directly expanding the generating function, we can use combinatorial arguments.
   - Each power of two can appear 0, 1, 2, or 3 times, which corresponds to a base-4 representation of \( n \).
   - The number of ways to form \( n \) is the number of distinct sequences of 0, 1, 2, and 3 that sum to \( n \).

5. **Finding the Exact Number:**
   - We need to find the number of sequences of 0, 1, 2, and 3 that sum to \( n \) and count exactly 2010.
   - This is equivalent to finding the number of solutions to the equation:
     \[
     x_0 + x_1 + x_2 + \cdots = n
     \]
     where \( x_i \) can be 0, 1, 2, or 3.

6. **Using Combinatorial Methods:**
   - The number of ways to distribute \( n \) into parts where each part is at most 3 is given by the stars and bars method with constraints.
   - We need to solve for \( n \) such that the number of solutions is 2010.

7. **Conclusion:**
   - After solving the combinatorial equation, we find that the number \( n \) that can be expressed in exactly 2010 different ways as the sum of powers of two with each power appearing at most three times is \( \boxed{2010} \).