A091318 -id:A091318 - cp's OEIS frontend

A091237 Lengths of runs in A039702.

1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 2, 3, 7, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 3, 4

Offset: 1

N. J. A. Sloane, Feb 22 2004

Lengths of runs, where a run is a succession of primes that are congruent mod 4.

In other words, RUNS transform of A039702. If the two initial 1's are omitted, this is the RUNS transform of A100672. - N. J. A. Sloane, Jan 11 2025

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A039702, A091318, A091267, A100672.

Most[Length /@ Split[Table[Mod[Prime[n], 4], {n, 200}]]] (* T. D. Noe, Sep 21 2012 *)

More terms from David Wasserman, Feb 28 2006

A091295 (Number of primes == 3 mod 4 less than 10^n) - (number of primes == 1 mod 4 less than 10^n).

Original entry on oeis.org

1, 2, 7, 10, 25, 147, 218, 446, 551, 5960, 14252, 63337, 118472, 183457, 951700, 3458334, 6284060, 2581691, 80743228, 259753425

Offset: 1

Enoch Haga, Feb 23 2004

			a(1) = 1 because below 10^1 3 and 7 are 3 mod 4 and 5 is 1 mod 4 and the difference is 2-1=1.

Hans Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Birkhauser, The distribution of primes between the two series 4n+1 and 4n+3, pages 73-77, with graphs.

Marc Deléglise, Pierre Dusart, and Xavier-Francois Roblot, Counting primes in residue classes, Math. Comp. 73 (2004), no. 247, 1565-1575.
Carlos Rivera, Puzzle 256, Jack Brennen old records, The Prime Puzzles & Problems Connection.

Cf. A091098, A091099. A002144, A002145, A091318, A091267.

a(n) = A091099(n) - A091098(n) = A093153(n) + 1. [Max Alekseyev, May 17 2009]

a(17)-a(20) from Marc Deleglise, Jun 28 2007

A145989 Run lengths of consecutive primes == 1 (mod 4) where the run length is at least 2.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 3, 2, 4, 2, 2, 2, 3, 3, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Enoch Haga, Oct 26 2008

The run lengths of 1's in A039702 are 1, 2, 1, 2, 1, 1, 1, 3,.. as listed in A091318. Deleting all 1's from A091318 generates this sequence here. - R. J. mathar, Sep 30 2011

The maximum run length in the first 1000 terms is 9. - Harvey P. Dale, Jul 27 2025

			a(1)=2 because this sequence includes consecutive runs of any length and this first occurrence > 1 is a run of 2.

Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007. Pp. 30-31. ISBN 978-1-885794-24-6

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055623, A054624, A145986, A145988 - A145994.

Length/@Select[Split[Table[If[Mod[p,4]==1,1,0],{p,Prime[Range[500]]}]],#[[1]]==1&&Length[#]>1&] (* Harvey P. Dale, Jul 27 2025 *)

10 'cluster primes
20 C=1
30 input "end #";L
40 for N=3 to L step 2
50 S=int(sqrt(N))
60 for A=3 to S step 2
70 B=N/A
80 if int(B)*A=N then cancel for:goto 170
90 next A
100 C=C+1
110 E=N/4:E=int(E):R=N-(4*E)
120 if R=1 then print N;:C1=C1+1:T1=T1+1:print T1
130 if R=3 then T1=0
140 if R=3 then print " ";N;:C3=C3+1:T2=T2+1:print T2
150 if R=1 then T2=0
160 if T1>10 or T2>10 then stop
170 next
180 print "Total primes=";C;:print "Type A";C1;"Type B";C3

A091267 Lengths of runs of 3's in A039702.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 7, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 5, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 3, 2, 2, 5, 5, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1, 3, 4, 1

Offset: 1

Enoch Haga, Feb 22 2004

Number of primes congruent to 3 mod 4 in sequence before interruption by a prime 1 mod 4.

			a(16)=4 because this is the sequence of primes congruent to 3 mod 4: 199, 211, 223, 227. The next prime is 229, a prime 1 mod 4.

Enoch Haga, Exploring prime numbers on your PC and the Internet with directions to prime number sites on the Internet, 2001, pages 30-31. ISBN 1-885794-17-7.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A091318, A039702, A091237.

t = Length /@ Split[Table[Mod[Prime[n], 4], {n, 2, 400}]]; Most[Transpose[Partition[t, 2]][[1]]] (* T. D. Noe, Sep 21 2012 *)

Count primes congruent to 3 mod 4 in sequence before interruption by a prime divided by 4 with remainder 1.

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

Enoch Haga, Mar 02 2004

			a(3)=31 because 31 single primes occur below 10^3, each interrupted in the run by a prime congruent to 3 mod 4.

Cf. A091318, A092637-A092665.

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 *)

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

Generate the prime sequence with primes labeled 1 mod 4 or 3 mod 4. Add count of primes to sequence if just one prime occurs before interruption by a prime congruent to 3 mod 4.

a(9) from Charles R Greathouse IV, Sep 30 2011

a(10)-a(11) from Chai Wah Wu, Mar 18 2018

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

Enoch Haga, Mar 02 2004

			a(3)=28 because 28 single primes occur below 10^3, each interrupted in the run by a prime congruent to 1 mod 4.

Cf. A091318, A092636 - A092665.

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 *)

Generate the prime sequence with primes labeled 1 mod 4 or 3 mod 4. Add count of primes to sequence if just one prime occurs before interruption by a prime congruent to 1 mod 4.

a(9)-a(11) from Chai Wah Wu, Mar 18 2018

cp's OEIS Frontend

A091237 Lengths of runs in A039702.

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A091295 (Number of primes == 3 mod 4 less than 10^n) - (number of primes == 1 mod 4 less than 10^n).

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A145989 Run lengths of consecutive primes == 1 (mod 4) where the run length is at least 2.

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UBASIC

A091267 Lengths of runs of 3's in A039702.

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A092636 Number of consecutive prime runs of 1 prime congruent to 1 mod 4 below 10^n.

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PARI

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A092637 Number of consecutive prime runs of 1 prime congruent to 3 mod 4 below 10^n.

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