In any (standard) positional number system, the number of symbols required to represent a number $n$ is asymptotic to $O(\text{ln}(n))$. For example, in the base 2 (binary), the number of symbols required to represent $n$ is $\text{floor}(\text{log}_2(n))+1$.
In constrast, in most tally mark systems, the number of symbols required to represent a number $n$ is asymptotic to $O(n)$. These systems are less efficient than the positional number systems since numbers require exponentially more symbols to represent.
My question
I want to know if there exist systems that are even more efficient than the positional number systems, perhaps on the cor even $O(1)$ level, which only have a finite number of symbols.
The only one I could find online is called factorial number system, but it requires an infinite number of symbols.
UPDATE: I also am open to non-positional systems, as long as they are below $O(\text{ln}(n))$ and use a finite collection of symbols.
Best Answer
If you have $c$ symbols and use $n$ of them there are only $c^n$ different strings, so that is the most numbers you can represent. That means to represent all the numbers up to $N$ you need $\log_c(N)$ symbols.