For the music application, Max Cooper uses items that appear in the OEIS catalogue pretty heavily. IMO, his works sound amazing. Tracks like Aleph 2 (https://vimeo.com/367747083) or Fibonacci Sequence (https://vimeo.com/991635293) stand out as immediate examples.

For something more mainstream even Tool lean into the Fibonacci sequence theme with Lateralus, almost to a meme level.

"how does music that relies on these weird sequences actually sound?"

Like random garbage, mostly. Music being music, someone can take that random garbage and try to make something that sounds good out of it. It is a common problem in music theory homework to be handed a more-or-less random melody and have to set a harmony to it. But the work there is done by the musician, not the random melody.

There is definitely some very common, but somewhat weird, belief that if we just feed pi into some sort of music algorithm that it will reveal some sort of hidden beauty of something or other. But pi is effectively a random number generator from this point of view. Nothing "pi"-ish comes out of such a mapping. You get more "pi" out of the story of the periodic sound itself than you do the sequence of sounds. Most OEIS sequences are also some combination of "effectively random" or "painfully boring" (e.g., a sequence of 10 ones).

On the music side, treating the 12 tones of the conventional scale as numbers 1 through 12 and then trying to map things to it is also somewhat weird and generally useless, because the musical relationship between the tones is quite non-trivial and anything but linear. In numeric sequences we generally want a 2 to strongly resemble a 3, but be quite distinct from a 14, but in music terms, two notes that are a semitone away from each other are often one of the more distant relationships. The diminished fourth, which is 6 away (e.g., a 1 and a 7) is probably the most distant, but the semitone is a strong contender for the next most distant. The strong relationships are the octave, 12 apart, then the fifth, then the fourth, and so on around the circle of fifths. So mapping integer sequences to musical notes in the straightforward and obvious way most people do is also generally not productive because whatever they are trying to draw out of the sequence of numbers is very unlikely to be "revealed" by mapping it naively to musical notes unless by a very bizarre (and probably deeply meaningful) coincidence the sequence of numbers has some sort of similar relationships as the 12-tone scale.

I would, however, be intrigued to see an example of someone mapping some mathematical object to music in some less trivial way that could actually reveal something interesting. Doesn't have to use tones. Maybe something that maps certain characteristics of something to the intensity of overtones or something. Or map something into an interesting polyrhythmic percussion line that somehow reflects something about the underlying sequence or mathematical object. Maybe there's something really interesting out there like this, but I expect it takes more than just shoving a sequence of numbers into the obvious integer mapping to conventional 12-tone pitches.

Perhaps the closest thing I've seen to this are the many videos on YouTube that map pitches to the order of operations of a sorting algorithm. Though I question if the audio brings anything new to the party that the visuals hadn't already brought.

I give an honorable mention to the people who are fairly aware that the constants are just random number generators and pit them against each other in Mario Party or similar: https://www.youtube.com/watch?v=zFcDgDXrrok I don't love the video but the idea is amusing.