# Gaps and density of numbers consisting of sums of products of primes where the number of factors for each prime is itself prime

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

• 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. – reuns Jan 21 '19 at 20:33
• 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. – phdmba7of12 Jan 21 '19 at 22:03
• as a test if it is clear, try to list the next three lowest numbers in the set – phdmba7of12 Jan 21 '19 at 22:09