Multiplicative digit sequences

One can consider generalizations of digit sequences in which numbers are broken into parts combined not by addition but by multiplication. Since numbers can be factored uniquely into products of powers of primes, a number can be specified by a list in which 1's appear at the positions of the appropriate Prime[m]^{n} (which can be sorted by size) and 0's appear elsewhere, as shown below. Note that unlike the case of ordinary additive digits, far more than Log[m] digits are required to specify a number m.