A few of those that pop up on the internet every now and then (for instance on [xkcd](https://xkcd.com/1047/) are actually found using [RIES](http://mrob.com/pub/ries/ries.php?target=3.1415), a program made for finding algebraic approximations given a decimal (possibly rounded) representation, by Robert Munafo.

This makes it easy to find such coincidences and many of those have been published. But these usually fall more under recreational or experimental math, as they are not appearing in the context of some "serious" research.