MathDB

For a positive integer

n

, a sum-friendly odd partition of

n

is a sequence

(a_1, a_2, \ldots, a_k)

of odd positive integers with

a_1 \le a_2 \le \cdots \le a_k

and

a_1 + a_2 + \cdots + 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} + \cdots + a_{i_r}

. (Two subsums

a_{i_1} + a_{i_2} + \cdots + a_{i_r}

and

a_{j_1} + a_{j_2} + \cdots + a_{j_s}

with

i_1 < i_2 < \cdots < i_r

and

j_1 < j_2 < \cdots < 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

Problem Statement