Consider a set of integers $Q$ such that the set of all positive integers $\mathbb{Z}$ is equivalent to the span of ever possible power tower
$$a_1^{a_2^{\ldots a_N}}$$ involving $a_i \in Q$.
In simpler terms. Take the integers, remove all square numbers, cube numbers, fourth powers, fifth powers, etc… And this remaining set is $Q$.
What is the density of $Q$ compared to positive $\mathbb{Z}$? Does it obey a theorem similar to the prime number theorem for primes? Are there infinity many numbers $x$, in $Q$ such that both $x$ and $2x$ are members of $Q$? Is there a formula for the elements of $Q$?
This is basically analogous to prime numbers except now it deals with exponents as opposed to multiplication.
Best Answer
The density of exponential primes (i.e. non-power numbers) is 1. In fact, there are so few perfect powers that the sum of the reciprocals of perfect powers converges: $$ \frac{1}{2^2} + \frac1{2^3} + \frac1{3^2} + \frac1{2^4} + \frac1{5^2} + ... \approx 0.87446... $$ For more details, see Wikipedia's article.