Given a set $A$ containing 10 positive integers, with the largest element denoted as $K$, we calculate all possible sums of elements from set $A$, including sums of 2, 3, 4, and so on, up to all 10 elements. These sums, along with the 10 numbers in set $A$, constitute multiset $B$.
If we require that there are no repeated elements within multiset $B$, what is the minimum value of $K$ among all sets $A$ meeting this requirement? and what is the corresponding set $A$?
This is sequence A276661 in the OEIS:
Least k such that there is a set S in {1, 2, ..., k} with n elements and the property that each of its subsets has a distinct sum.
The known values for $n\in\{0,1,\dots,9\}$ are
0, 1, 2, 4, 7, 13, 24, 44, 84, 161
An upper bound for $n=10$ is $309$:
{148,225,265,285,296,302,305,307,308,309}
My intuition implies that you start with 1, and then you add 1 to it to get the second number, resulting in 2. The third number will be the sum of all the former numbers, plus one, so it will be 3. This pattern continues repeatedly, resulting in the sequence:
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024]
Note that this sequence can be represented as powers of 2: $2^n$, where $n$ is the position in the sequence.
Now, this makes sense since that is the precise way we represent numbers in the base-2 system. The minimum number which cannot be represented by the set of bits {1, $\ldots$, n} (or any subset of it) is $2^{n+1}$.
