Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of natural numbers all raised to *different* powers?
To make this question interesting, one has to impose a few additional restrictions:
- If first powers were allowed, every natural number $n$ could be trivially represented as $n=n^1$, so all exponents must be greater than $1$
- If $1$ were allowed as a base, every natural number $n$ could be trivially represented as $n=1^{a_1}+...+1^{a_n}$, so the natural numbers summed over must all be greater than $1$
Example: $12=2^2+2^3$ (smallest number representable under the above rules that is not a perfect power)