--description--

Let $S(A)$ represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true:

1. $S(B) ≠ S(C)$; that is, sums of subsets cannot be equal.
2. If B contains more elements than C then $S(B) > S(C)$.

If $S(A)$ is minimised for a given n, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.

$$\begin{align} & n = 1: \{1\} \\ & n = 2: \{1, 2\} \\ & n = 3: \{2, 3, 4\} \\ & n = 4: \{3, 5, 6, 7\} \\ & n = 5: \{6, 9, 11, 12, 13\} \\ \end{align}$$

It seems that for a given optimum set, $A = \{a_1, a_2, \ldots, a_n\}$, the next optimum set is of the form $B = \{b, a_1 + b, a_2 + b, \ldots, a_n + b\}$, where b is the "middle" element on the previous row.

By applying this "rule" we would expect the optimum set for $n = 6$ to be $A = \{11, 17, 20, 22, 23, 24\}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = \{11, 18, 19, 20, 22, 25\}$, with $S(A) = 115$ and corresponding set string: 111819202225.

Given that A is an optimum special sum set for $n = 7$, find its set string.

Note: This problem is related to Problem 105 and Problem 106.

--hints--

optimumSpecialSumSet() should return the string 20313839404245.

assert.strictEqual(optimumSpecialSumSet(), '20313839404245');

--seed--

--seed-contents--

function optimumSpecialSumSet() {

  return true;
}

optimumSpecialSumSet();

--solutions--

// solution required