My favorite is this FRACTRAN program: (17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 1/17, 11/13, 13/11, 15/2, 1/7, 55/1) with input 2.
It doesn't quite produce primes directly. It gives a sequence of integers, some of which are powers of 2. Those powers of 2 are 2^2, 2^3, 2^5, 2^7, 2^11, ..., i.e., the sequence of 2 to the power of primes.
For those who have never seen FRACTRAN it is an esoteric programming language invented by John Conway. It is Turing-complete.
Here's how you execute a FRACTRAN program, with input N.
1. Step through the list of fractions in order until you find a fraction f such that fN is an integer or you run out of fractions.
It doesn't quite produce primes directly. It gives a sequence of integers, some of which are powers of 2. Those powers of 2 are 2^2, 2^3, 2^5, 2^7, 2^11, ..., i.e., the sequence of 2 to the power of primes.
For those who have never seen FRACTRAN it is an esoteric programming language invented by John Conway. It is Turing-complete.
Here's how you execute a FRACTRAN program, with input N.
1. Step through the list of fractions in order until you find a fraction f such that fN is an integer or you run out of fractions.
2. If you run out of fractions halt.
3. Output fN.
4. Replace N with fN.
5. Goto step 1.
Wikipedia gives some sample programs and explains in detail how the heck FRACTRAN can compute: https://en.wikipedia.org/wiki/FRACTRAN