$2^3$ and $3^2$ are close together; $11^2$, $5^3$, and $2^7$ (121, 125, and 128) are close together; $3^5$, $2^8$, and maybe $17^2$ (243, 256, and 289) are close together. $7^3$ is close to $19^2$ (343 and 361). $3^7$ is very nearly $13^3$ (2187 and 2197), which is very nearly $47^2$ (2209). $19^4$ is close to $2^{17}$ (130,321 and 131,072). Further examples are easy to find.
One might expect coincidences to get farther apart and rarer as the numbers get larger, but that doesn't seem to happen. For example, $13^{11}$ and $23^9$ differ by about one part in 200 (1,792,160,394,037 and 1,801,152,661,463).
Is this all just the law of small numbers at work? How many such coincidences would one naïvely expect? Is there any evidence that the number of such coincidences is, or is not, more than one would naïvely expect? Is anything known about the distribution of prime powers?
