I'll be more systematic.
We know that no prime after 2 is divisible by 2, so we should look in odd numbers:
2k + 1
We now try excluding everything divisible by 3:
6k + {1, 5}
Number of add-on numbers: (2)
Now everything divisible by 5:
30k + {1, 7, 11, 13, 17, 19, 23, 29}
(8)
Now everything divisible by 7:
210k + {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209}
(48)
Some of these add-on numbers are composite: 121=112, 143=11*13, 169=132, 187=11*17, 209=11*19
I recall an Isaac Asimov science essay from long ago where he proposed that numbers of form 6k+1 and 6k+5 would be good for studying the distribution of primes, because they filter out many composite numbers without being overly complicated.