"subprimes" numbers

[Edgard Neuman]

Hi,

Here is a math question : 
First I'm going to define some things (some  names may already exists that I don't know of, so please take my definition into consideration)

- let's call p[n] the nth-rank prime number p[0]=1, p[1]=2, p[2]=3, p[3]=5 etc
- as you know, each integer >0 can be written as a product of integer powers of prime numbers.. let's call it the "prime writing" of a number... i'll write u[n]
so for any integer X we have
X = product( p[n]  ^ u[n] ) 
- we can extend this to rational numbers, simply by allowing u[n] <0

My question is : can we define a set of irrational numbers in ]0 ; 1[  that extends p[n] when n<0 and are the building blocks for irrational numbers  ? Let's call them subprimes..
Those numbers would have the properties following :
- they are not power/products of primes and other sub-primes and of course integer powers of some other real number   (other than themselves) 

Are they already known ? Do they exist ? How to construct them ? 
I have some (very faint) clue : 
When you elevate these numbers to positive powers , you get closer and closer to 0.. so the more you go close to 0, the more likely to find a power of a bigger subprime.. so the density must decrease closer to 0.. you get some sort of sieve, but closer and closer to 0.

Edited October 21, 2019 by Edgard Neuman

[Edgard Neuman]

I realize maybe we would have to define them each as a unique "set" of integer powers of a specific irrational number between ]0;1[  but the idea remains the same

Edited October 21, 2019 by Edgard Neuman