Of course there' s a sequence but it doesn't seem to have any simple mathematical formula. So, neither do most real world phenomena without bringing complex numbers into it. Simply put, every number that is not prime is so many along the infinite integer list from the start. Every second, third, fourth, fifth, sixth and so on is not prime except the first of its kind unless it has already been counted out. So the fourth and sixth have already been eliminated in the seconds, and the ninth in the thirds.
It simplifies matters a little that all multiples of odd numbers are odd, so a possible prime must be 2 higher.I'm trying to think of a generalised heuristic starting with low numbers. 3×5=15, 3×7=21. There are two odd numbers between them, so they must be prime. 5×11=55, 5×11=65. Eliminate the numbers between for divisibility by lower primes 2 and 3, or start by generating all non-primes and see what is left out. Primes are a kind of inverse to Lowest Common Denominators. There is a relationship to distance between them and any possible divisors, though it is not obvious. Possibly, if the distance between them is not prime, then they are, so one possiblity of generating primes is to add the multiple of some lower primes to a highest known prime. Just which isn't obvious even at low values.
