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%I #17 Jun 09 2019 21:16:21
%S 0,2,14,116,780,6066,49510,417230,3631524,32070665,287366058
%N Number of consecutive prime runs of 2 primes congruent to 1 mod 4 below 10^n.
%F Generate the prime sequence with primes labeled 1 mod 4 or 3 mod 4. Add count of primes to sequence if just 2 primes occur before interruption by a prime congruent to 3 mod 4.
%e a(4)=116 because 116 pairs of primes occur below 10^4, each run interrupted by a prime congruent to 3 mod 4.
%t A002145 = Join[{0}, Select[4 Range[0, 10^4] + 3, PrimeQ[#] &]];
%t A002144 = Select[4 Range[0, 10^4] + 1, PrimeQ[#] &];
%t lst = {}; Do[If[Length[s = Select[A002144, Between[{A002145[[i]], A002145[[i + 1]]}]]] == 2, AppendTo[lst, Last[s]]], {i, Length[A002145] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}] (* _Robert Price_, Jun 09 2019 *)
%o (PARI) ispRun(p1)={ local(p2,p3) ; if(!isprime(p1) || (p1 %4 ==3) || (precprime(p1-1) % 4 ==1), return(0), p2=nextprime(p1+1) ; if( p2 %4 == 3, return(0), p3=nextprime(p2+1) ; if( p3 %4 == 3, return(1), return(0) ) ; ) ; ) ; } { an=0 ; n=1 ; p=prime(1) ; while(1, if( (p<10^n) && (nextprime(p+1) >= 10^n), print(an); n++ ; ) ; an += ispRun(p) ; p=nextprime(p+1) ; ) } \\ _R. J. Mathar_, Sep 25 2006
%Y Cf. A092640, A092641.
%K more,nonn
%O 1,2
%A _Enoch Haga_, Mar 02 2004
%E a(9)-a(11) from _Chai Wah Wu_, Mar 18 2018
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