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A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.

%I A093183 #23 May 30 2019 13:45:02
%S A093183 0,3,74,1114,13437,151311,1642197,17405273,181925434,1883327626,
%T A093183 19364371468,198115934511,2019328584101
%N A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.
%C A093183 Split the odd nonprime sequence A014076 into two subsequences A091113 and A091236 with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 1 mod 4 occurs before interruption of a nonprime congruent to 3 mod 4.
%C A093183 Otherwise said: count the nonprimes congruent to 1 mod 4 such that the next larger and next smaller odd nonprime is congruent to 3 mod 4. - _M. F. Hasler_, Sep 30 2018
%e A093183 a(3) = 74 because 74 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4.
%e A093183 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
%p A093183 A014076 := proc(n)
%p A093183     option remember;
%p A093183     if n = 1 then
%p A093183         1;
%p A093183     else
%p A093183         for a from procname(n-1)+2 by 2 do
%p A093183             if not isprime(a) then
%p A093183                 return a;
%p A093183             end if;
%p A093183         end do:
%p A093183     end if;
%p A093183 end proc:
%p A093183 isA091113 := proc(n)
%p A093183     option remember;
%p A093183     if modp(n,4) = 1 and not isprime(n) then
%p A093183         true;
%p A093183     else
%p A093183         false;
%p A093183     end if;
%p A093183 end proc:
%p A093183 isA091236 := proc(n)
%p A093183     option remember;
%p A093183     if modp(n,4) = 3 and not isprime(n) then
%p A093183         true;
%p A093183     else
%p A093183         false;
%p A093183     end if;
%p A093183 end proc:
%p A093183 ct := 0 :
%p A093183 n := 1 :
%p A093183 for i from 2 do
%p A093183     odnpr := A014076(i) ;
%p A093183     prev := A014076(i-1) ;
%p A093183     nxt := A014076(i+1) ;
%p A093183     if isA091113(odnpr) and isA091236(prev) and isA091236(nxt) then
%p A093183         ct := ct+1 ;
%p A093183     end if;
%p A093183     if odnpr< 10^n and nxt >= 10^n then
%p A093183         print(n,ct) ;
%p A093183         n := n+1 ;
%p A093183     end if;
%p A093183 end do: # _R. J. Mathar_, Oct 02 2018
%t A093183 A091113 = Select[4 Range[0, 10^5] + 1, ! PrimeQ[#] &];
%t A093183 A091236 = Select[4 Range[0, 10^5] + 3, ! PrimeQ[#] &];
%t A093183 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 *)
%Y A093183 Cf. A091113, A091236, A093184-A093188, A093397-A093399, A092636.
%K A093183 more,nonn
%O A093183 1,2
%A A093183 _Enoch Haga_, Mar 30 2004
%E A093183 a(9)-a(13) from _Bert Dobbelaere_, Dec 19 2018