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) = 74 because 74 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4. Below 10^2 = 100, there are only a(2) = 3 isolated odd nonprimes congruent to 1 mod 4: 33, 57 and 93. (Credits: _Peter Munn_, SeqFan list.) - _M. F. Hasler_, Sep 30 2018
A014076 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do: end if; end proc: isA091113 := proc(n) option remember; if modp(n,4) = 1 and not isprime(n) then true; else false; end if; end proc: isA091236 := proc(n) option remember; if modp(n,4) = 3 and not isprime(n) then true; else false; end if; end proc: ct := 0 : n := 1 : for i from 2 do odnpr := A014076(i) ; prev := A014076(i-1) ; nxt := A014076(i+1) ; if isA091113(odnpr) and isA091236(prev) and isA091236(nxt) then ct := ct+1 ; end if; if odnpr< 10^n and nxt >= 10^n then print(n,ct) ; n := n+1 ; end if; end do: # R. J. Mathar, Oct 02 2018
A091113 = Select[4 Range[0, 10^5] + 1, ! PrimeQ[#] &]; A091236 = Select[4 Range[0, 10^5] + 3, ! PrimeQ[#] &]; lst = {}; Do[If[Length[s = Select[A091113,Between[{A091236[[i]], A091236[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091236] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 5}] (* 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)=28 because 28 single primes occur below 10^3, each interrupted in the run by a prime congruent to 1 mod 4.
A002144 = Join[{0}, Select[4 Range[0, 10^4] + 1, PrimeQ[#] &]]; A002145 = Select[4 Range[0, 10^4] + 3, PrimeQ[#] &]; lst = {}; Do[If[Length[s = Select[A002145, Between[{A002144[[i]], A002144[[i + 1]]}]]] == 1, AppendTo[lst, Last[s]]], {i, Length[A002144] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}] (* Robert Price, May 31 2019 *)
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