What is the name (if there is one) of the "full factorization representation" of a number, in which also the powers of the factors are (recursively) decomposed until all the numbers used in the representation are primes?
For example: instead of writing: $n = 2^{4} 3 ^{98304}$, write:
$$n = 2^{2^2 } 3 ^{2^{3 \cdot 5} \cdot 3}$$
($4 = 2^2$, $98304 = 2^{15} \cdot 3$, $15 = 3 \cdot 5$)
I think it is known as Prime Power Factor Decomposition
Wiki article on Arithmetic Functions