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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.
EXAMPLE
a(5)=124 because 124 sets of 4 primes occur below 10^5, each run interrupted by a prime congruent to 3 mod 4.
MATHEMATICA
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 *)
PROG
(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