A092636
Number of consecutive prime runs of 1 prime congruent to 1 mod 4 below 10^n.
Original entry on oeis.org
1, 5, 31, 208, 1555, 12465, 102704, 869060, 7540342, 66571720, 595513442
Offset: 1
a(3)=31 because 31 single primes occur below 10^3, each interrupted in the run by a prime congruent to 3 mod 4.
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A002144 = Select[4 Range[0, 10^4] + 1, PrimeQ[#] &];
A002145 = Select[4 Range[0, 10^4] + 3, PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A002144, Between[{A002145[[i]], A002145[[i + 1]]}]]] == 1, AppendTo[lst, Last[s]]], {i, Length[A002145] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}] (* Robert Price, May 31 2019 *)
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a(n)=my(p=2,q=3,t);forprime(r=5,nextprime(10^n),if(q%4==1&&p%4==3&&r%4==3,t++);p=q;q=r);t \\ Charles R Greathouse IV, Sep 30 2011
A092637
Number of consecutive prime runs of 1 prime congruent to 3 mod 4 below 10^n.
Original entry on oeis.org
1, 3, 28, 217, 1570, 12515, 102942, 867677, 7541800, 66571277, 595524791
Offset: 1
a(3)=28 because 28 single primes occur below 10^3, each interrupted in the run by a prime congruent to 1 mod 4.
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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 *)
A092663
Number of consecutive prime runs of 10 primes congruent to 1 mod 4 below 10^n.
Original entry on oeis.org
0, 0, 0, 0, 0, 5, 19, 323, 3653, 37544, 381413, 3799344, 37591054
Offset: 1
a(6)=5 because 5 sets of 10 primes occur below 10^6, each run interrupted by a prime congruent to 3 mod 4. These runs start at prime(31798)=373649, prime(41181)=495377, at prime(42241)=509389, at prime(50017)=612109, *not* at prime(61457) which has a larger run length, and at prime(63146)=789097.
A092664
Number of consecutive prime runs of 10 primes congruent to 3 mod 4 below 10^n.
Original entry on oeis.org
0, 0, 0, 0, 0, 3, 23, 337, 3605, 38037, 380962, 3799462, 37594962
Offset: 1
a(6)=3 because 3 sets of 10 primes occur below 10^6, each run interrupted by a prime congruent to 1 mod 4.
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