The largest integer that is not a sum of distinct powers?
http://primes.utm.edu/curios/page.php/23.html
To find the largest integer that is not a sum of distinct powers, we can use a simple approach.
First, let's consider the powers of 2. The powers of 2 are 1, 2, 4, 8, 16, 32, and so on. Each power of 2 is a distinct integer, and any sum of distinct powers of 2 will also be a distinct integer.
So, the smallest integer that cannot be represented as a sum of distinct powers of 2 is 3, because it cannot be expressed as 2 + 1.
Next, let's consider the powers of 3. The powers of 3 are 1, 3, 9, 27, 81, and so on. Again, each power of 3 is a distinct integer, and any sum of distinct powers of 3 will also be a distinct integer.
The smallest integer that cannot be represented as a sum of distinct powers of 3 is 2, because it cannot be expressed as 1 + 1.
Continuing this pattern, we can consider the powers of 4, 5, 6, and so on. For each power, we can find the smallest integer that cannot be represented as a sum of distinct powers.
The largest integer that cannot be represented as a sum of distinct powers is the maximum of these values. Let's go through a few powers to illustrate this process:
- For powers of 2, the largest integer that cannot be represented as a sum of distinct powers is 3.
- For powers of 3, the largest integer that cannot be represented as a sum of distinct powers is 2.
- For powers of 4, the largest integer that cannot be represented as a sum of distinct powers is 7 (because it cannot be represented as 4 + 2 + 1).
- For powers of 5, the largest integer that cannot be represented as a sum of distinct powers is 1.
By continuing this process, we find that the largest integer that cannot be represented as a sum of distinct powers of any integer is 15.
Therefore, the largest integer that is not a sum of distinct powers is 15.