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(2)=2 because in A092636 the count is 5 and in A092637 the count is 3. 5-3=2.
a(3)=76 because 76 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4
A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &]; A091236 = Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &]; lst = {}; Do[If[Length[s = Select[A091236, Between[{A091113[[i]], A091113[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091113] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 4}] (* Robert Price, May 30 2019 *)
a(3)=2 because in A093183 the count is 74 and in A093184 the count is 76. 76-74=2.
a(3)=31 because 31 single primes occur below 10^3, each interrupted in the run by a prime congruent to 3 mod 4.
A002144 = Select[4 Range[0, 10^4] + 1, PrimeQ[#] &]; A002145 = Select[4 Range[0, 10^4] + 3, PrimeQ[#] &]; lst = {}; Do[If[Length[s = Select[A002144, Between[{A002145[[i]], A002145[[i + 1]]}]]] == 1, AppendTo[lst, Last[s]]], {i, Length[A002145] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}] (* Robert Price, May 31 2019 *)
a(n)=my(p=2,q=3,t);forprime(r=5,nextprime(10^n),if(q%4==1&&p%4==3&&r%4==3,t++);p=q;q=r);t \\ Charles R Greathouse IV, Sep 30 2011