There is no limit to this! The Green Tao theorem tells you that the sequence of prime numbers contains arbitrarily long arithmetic progressions. This means that using $a+b\cdot n$ you can get as many primes as you want for some $a,b$ and consecutive values of $n$. But the theorem does not tell you how to find $a,b$. The longest known such sequence can be ...


The function with rule produces distinct primes for $n$ up to $25$. For proof, see the third bullet point on this list of prime number records. It is valid for $x=0,1,...,23$, so I substitute $n=x+2$ so that the set of valid inputs begins at $2$. The function is clearly strictly increasing and so the primes must be distinct. It is apparently the longest ...


How about this


Perhaps there is a slight trick to this one


For $ n = 2, 3, 4, 5 $, the function produces distinct prime numbers:


A solution (probably many more of them exist):


If I understood everything correctly, this solution is also acceptable. If not - it's still very interesting and simple one. It's valid for $n=0,1,2\dots,28$


Another solution is: The solutions correspond to:


A partial answer based on both my own's answer to the original question and Steve's comments on it:


I’m going to guess:

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