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%I #25 Mar 18 2018 17:25:39
%S 0,0,2,10,124,1047,8756,78845,703152,6387276,58448789
%N Number of consecutive prime runs of just 4 primes congruent to 1 mod 4 below 10^n.
%C Conjecture: a(n) ~ 10^n / (64 log 10 * n). - _Charles R Greathouse IV_, Oct 24 2011
%F Generate the prime sequence with primes labeled 1 mod 4 or 3 mod 4. Add count of primes to sequence if just 4 primes occur before interruption by a prime congruent to 3 mod 4.
%e a(5)=124 because 124 sets of 4 primes occur below 10^5, each run interrupted by a prime congruent to 3 mod 4.
%t p1 = p3 = 0; p = 15; s = Mod[{2, 3, 5, 7, 11, 13}, 4]; Do[ While[p < 10^n, If[s == {3, 1, 1, 1, 1, 3}, p1++]; If[s == {1, 3, 3, 3, 3, 1}, p3++]; p = NextPrime@ p; s = Join[ Take[s, -5], {Mod[p, 4]}]]; Print[{p1, p3}], {n, 2, 9}] (* _Robert G. Wilson v_, Sep 30 2011 *)
%o (PARI) a(n)=my(s,t);forprime(p=2,nextprime(10^n),if(p%4==1,t++,s+=t==4;t=0));s \\ _Charles R Greathouse IV_, Oct 24 2011
%Y Cf. A092646, A092647.
%K more,nonn
%O 1,3
%A _Enoch Haga_, Mar 02 2004
%E a(11) from _Chai Wah Wu_, Mar 18 2018
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