That article made me think of another kind of counting problem:
How many ways do you get a prime number by changing one zero bit in a particular prime number into a one?
17 in binary (most significant bit first) is 10001; when you you change that to 10011 you have 19, but that's it, any other zero bit changed to one is results in a composite number: 10101 is 21, 11001 is 25; so the answer for 17 is 1 (unless you keep going with higher powers of 2; is there a limit? 17+8192=8209 another prime). NB amazingly adding powers of 2 to 17 produces a few unexpected perfect squares: 17+8=25, 17+32=49, 17+64=81, 17+512=529
The basic problem is finding how many primes you can generate from another prime by adding powers of 2, skipping those powers of 2 that already occur in the additive decomposition into powers of 2 of the prime.