I am tackling Project Euler's [problem number 88](http://projecteuler.net/problem=88), which in a nutshell reads:

> 
> Let $S\_n$ be the set of sequences of natural numbers $(s\_1,s\_2,...,s\_n)$ where $s\_1\leqslant s\_2\leqslant\cdots\leqslant s\_n$ and
>  $$
> \sum s\_i=\prod s\_i.\tag{\*}
> $$
>  Observe that
>  $$\begin{align}
> S\_2&=\{(2,2)\},\\
> S\_3&=\{(1,2,3)\},\\
> S\_4&=\{(1,1,2,4)\},\\
> S\_5&=\{(1,1,1,2,5),(1,1,1,3,3),(1,1,2,2,2)\},\quad\text{and}\\
> S\_6&=\{(1,1,1,1,2,6)\}.
> \end{align}$$
>  Moreover, let $\text{min }S\_n$ be the smallest sum of the elements of the sequences in this set. For example,
>  $$
> \text{min }S\_5=8.
> $$
>  **What is the sum of the elements of**
>  $$
> \bigcup\_{i=2}^{12000}\{\text{min }S\_i\}?
> $$
>  If $6$ replaced $12000$ above, the answer would be $30$.
> 
> 
> 

I conjectured that an element of a sequence in $S\_n$ is no greater than $n$, and obtained $S\_n$ by computing all natural-number sequences $s\_1\leqslant s\_2\leqslant\cdots\leqslant s\_n\leqslant n$ and picking those that satisfy $(\*)$.

The program runs perfectly for small $n$. However, when I tried to answer the boldface question, my program ran for hours with no result... and kept going. I shut it down when I discovered that there are

$$
\binom{23999}{11999}=\text{number with }7223\text{ digits}
$$

natural-number sequences $s\_1\leqslant s\_2\leqslant\cdots\leqslant s\_{12000}\leqslant12000$.

Since I cannot wait until the end of the universe, could someone give me a better alternative on how to approach this problem? Could I perhaps halt computing sequences when a certain condition is met?

**Edit 1**: I just conjectured that $\text{sum }S\_n\leqslant2n$. I will now use this to crop my number of brute-force sequences.

**Edit 2**: The speed improved, but it was not enough...