, 2013

For a positive integer $n$, a sum-friendly odd partition of $n$ is a sequence $(a_1, a_2, \dots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \dots \le a_k$ and $a_1 + a_2 + \dots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be uniquely written as a subsum $m = a_{i_1} + a_{i_2} + \dots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \dots + a_{i_r}$, and $a_{j_1} + a_{j_2} + \dots + a_{j_s}$ with $i_1 < i_2 < \dots < i_r$ and $j_1 < j_2 < \dots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.