This was inspired by the following paper:

- J. Arias de Reyna, J. van de Lune,
*"How many $1$s are needed?" revisited*, arXiv link.

It might help explain my question better, because my question is actually similar to those in the problem.

So, I was curious, what if we were only allowed to use the operations $+,!$.

Using the same notation as the paper, $$\|a\|$$ is the least number of 1's used to represent $a$ using the provided (above) symbols (and parentheses). If my explanation is not great, here are some values of $\|a\|$.

- $\|1\|=\|1\|=1$
- $\|2\|=\|1+1\|=2$
- $\|3\|=\|1+1+1\|=3$
- $\|6\|=\|(1+1+1)!\|=3$
- $\|121\|=\|(1+1+1+1+1)!+1\|=6$

What would be a time-efficient means of calculating $\| a \|$ for large $a$?