This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3)=5 because in A093187 the count is 10 and in A093188 the count is 5. 10-5=5.
a(5)=305 because 305 sets of 3 primes occur below 10^5, each run interrupted by a prime congruent to 3 mod 4.
A002145 = Join[{0}, Select[4 Range[0, 10^4] + 3, PrimeQ[#] &]]; A002144 = Select[4 Range[0, 10^4] + 1, PrimeQ[#] &]; lst = {}; Do[If[Length[s = Select[A002144, Between[{A002145[[i]], A002145[[i + 1]]}]]] == 3, AppendTo[lst, Last[s]]], {i, Length[A002145] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}] (* Robert Price, Jun 09 2019 *)
a(5)=323 because 323 sets of 3 primes occur below 10^5, each run interrupted by a prime congruent to 1 mod 4.