The distribution of powers of primes

Question

How often do we see two or more powers of primes between two consecutive primes $p_k$ and $p_{k+1}$? One example is $p_4=7$ and $p_5=11$; we have $$ 7 < 2^3 < 3^2 < 11. $$ Are there any other examples of this kind?

Answer 1

Let $p$ denote a prime. We call the integer $p^a, \ a\in{\mathbb N},$ a prime power.

In the following, assume $a>1$. Prime powers with exponents $a>1$ are more rare than primes. It is not difficult to see that the vast majority of intervals $[p_k,p_{k+1}]$ do not contain any prime powers. Only a zero proportion of such intervals contain a prime power (even though there are infinitely many prime powers). Examples like $$7<2^3<3^2<11 \qquad \mbox{and} \qquad 23<5^2<3^3<29,$$ with more than one prime power in a single interval $[p_k,p_{k+1}]$, are extremely rare. Moreover, on probabilistic grounds we should expect that the total number of examples like this is finite. (I doubt that there is a rigorous proof of this. This result seems to be no easier than the Mihailescu theorem a.k.a. Catalan conjecture.)

Some numerical data. Among the first 10000 prime powers, there are only five examples: $$ 7 < 8=2^3 < 9=3^2 < 11 \\ 23 < 25=5^2 < 27=3^3 < 29 \\ 113 < 121=11^2 < 125=5^3 < 127 \\ 2179 < 2187=3^7 < 2197=13^3 < 2203 \\ 32749 < 32761=181^2 < 32768=2^{15} < 32771 $$

According to OEIS (https://oeis.org/A053706) there are no other examples for primes up to $2^{63}$. Thanks Matthew Conroy for pointing out the OEIS sequence!