Product-Over-Integers Expansion Of Real

[Leroy Quet]

Let A be a strictly increasing sequence of positive integers

where

sum{k=elements of A}  1/k

diverges.


For x = any real  > 1,

we can find an infinite number of expansions (each a sub-sequence of A)

such that:

x = product{k=some elements of A}  1/(1-1/k)

But we can find the specific "greedy-algorithm" subsequence, where each 
integer is chosen so that the partial-product remains <= x, and each new 
term is the lowest unpicked element of A such that the partial product is 
<= x.
(Such an expansion is, obviously, infinite if x is irrational.)

For example, either if A consists of all positive integers >= 2 or 
consists of just the primes, then, for x = pi, we have the expansion's 
terms

2, 3, 23,...

(pi = 1/(1-1/2) *1/(1-1/3) *1/(1-1/23) *...,
if I calculated right.)


I guess we can also ask about the subsequences of A such that

x = product{k=some elements of A} (1 + 1/k).


I bet this idea is not new. Are any such expansions already in the 
Encyclopedia of Integer Sequences?

Is there anything else known about such expansions?

thanks,
Leroy Quet