A couple of years ago, I stumbled across a nice little site called The On-Line Encyclopedia of Integer Sequences at http://oeis.org/Seis.html. If a sequence has turned up anywhere in the mathematical literature, you can find it here, along with its definition, additional terms, relationship to other sequences, and even an option to listen to the sequence as a musical tune (check out the Fibonacci sequence A000045).
If you feed it the numbers up through 60 in the puzzler, it gives you back four sequences, A014574 (Average of twin-prime pairs), A129297 (Nonnegative integers m such that m^2-1 has no divisors d with 1<d<m-1, which also has the numbers 0,1,2,3 at the beginning), A072570 (Even interprimes i = (p+q)/2 (where p, q are consecutive primes) such that (q-p)/2 is not divisible by 3, which is further described as a superset of A014574) and A167777 (Even single (or even isolated) numbers, which also includes the number 2 at the beginning).
All four of these, including the ones with extra terms at the beginning, continue with the same numbers for some time, so whichever one is being used the missing number in the Puzzler is the same.
(The sequence A133377 is one I came up with, but someone else actually submitted it.)