Instead of e, I have tried other constants such as pi, but it doesn't look as good.
I guess there is another constant that makes the distribution look even nicer.
I don't think this is anything special about primes or e - if you replace prime(n+1) with just (n+1) itself you get the same sort of patterns. But it is something to do with approximations of irrationals by rationals - you might want to look into continued fractions. Try replacing e with a rational number a/b (say 8/3 or 11/4); then you get b horizontal-ish lines, corresponding to the different remainders of n when divided by b. So the pattern you get with pi isn't "as good" because pi is famously close to 22/7.
The primes are somewhat evenly spaced with this transformation, I'm the author of it.
A342730: a(n) = floor((frac(e * n) + 1) * prime(n+1)).
https://oeis.org/A342730/a342730.png
Instead of e, I have tried other constants such as pi, but it doesn't look as good. I guess there is another constant that makes the distribution look even nicer.