Representing a Number as the Sum of Powers of Form $k^k$.

Question

So I was wondering if it would be useful to instead of writing a number in base $2$ or $3$, we use functions in general as bases. So like writing it as the sum of squares or other increasing functions. So for a number $N$, its base $f(n)$ representation could be found basically in the same way that you would find the base $2$ representation of $N$. The representation could be written as a "string" of characters and each character could be written in its binary representation. For example, $8153 = [1000][1][101][11]$ which took $10$ bits to write out. I wanted to see if there was a function that would dramatically reduce the number of bits required to store a number.