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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

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    $\begingroup$ I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $\lceil k \log k\rceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot. $\endgroup$ – reuns Jan 21 '19 at 20:33
  • $\begingroup$ to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept. $\endgroup$ – phdmba7of12 Jan 21 '19 at 22:03
  • $\begingroup$ as a test if it is clear, try to list the next three lowest numbers in the set $\endgroup$ – phdmba7of12 Jan 21 '19 at 22:09

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