For the following set of numbers: $$ \{ n \} =\sum_{i=1}^{\infty} b_i p_i^{p_{j_i}} $$ where each $b_i$ (b for binary) is either 1 or zero
each $n$ in the set $\{ n \}$ has a unique set of $\{b_i\}$ and $\{j_i\}$
Is there an anticipated asymptotic density for all possible $n$ thus defined?
first several lowest such $n$ start as 4, 8, 9, 13, 17, 25, 27, 29, 31, 32, 33...
this is $2^2,2^3,3^2,3^2+2^2,3^2+2^3,5^2, 3^3, 5^2+2^2, 3^3+2^2, 2^5, 5^2+2^3$, etc