Prime Factors

I just found this post lurking unpublished. I like it so let's let it out into the world.

So I wondered what would happen if instead of using base 10, each place in a number represented a prime factor. The "units" is instead the number of times each number is multiplied by 1 (this is always 1 for every number equal or greater than 1). The "tens" becomes the number of times 2 features as a prime factor in a number. So 11 in this prime factor number system is 2 in decimal. 21 is 4 in decimal as 4 = 2^2 * 1. The "hundreds" is instead the number of prime factor 3s. Is there a name for this system and is it used for anything? I can't be the first person to have thought of this.

Decimal Number	Prime Factor Number	Name
1	1	Root
2	11	Octave
3	101	Fifth
4	21	Octave
5	1001	Third
6	111	Fifth
7	10001	septimal (not seventh)
8	31	Octave
9	201	Second
10	1011	Third
11	100001	Tritone?)
12	121	Fifth
13	1000001	(Minor Sixth?)
14	10011	septimal
15	1101	Major Seventh
16	41	Octave