A145986 -id:A145986 - cp's OEIS frontend

A145994 Last prime in a run of at least 2 consecutive primes of the form 4k+3.

11, 23, 47, 71, 83, 107, 131, 167, 227, 311, 367, 383, 443, 503, 631, 647, 691, 727, 751, 827, 863, 919, 971, 991, 1091, 1171, 1283, 1319, 1427, 1451, 1471, 1487, 1543, 1583, 1667, 1787, 1847, 1871, 1987, 2011, 2087, 2111, 2207, 2267, 2351, 2411, 2467, 2543, 2591, 2671, 2687

Offset: 1

Enoch Haga, Oct 26 2008

			a(1)=11 because this sequence includes consecutive runs of any length >1 and this ending term in a run of 2 is 11.

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

Cf. A039702, A055623, A145986, A145988, A145990, A145991, A145992 (run lengths), A145993 (first prime in run)

A145994 := proc()
    local m,p,r,i,lp ;
    m := 3 ;
    p := 2 ;
    r := 0 ;
    for i from 2 to 1000 do
        if modp(p,4) = m then
            r := r+1 ;
        else
            if r > 1 then
                printf("%d,",prevprime(p)) ;
            end if;
            r := 0;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145994() ; # R. J. Mathar, Aug 29 2018

Last /@ Select[Split[Select[4Range[1000]+3, PrimeQ], #2 == NextPrime[#1]&], Length[#]>1&] (* Jean-François Alcover, Mar 26 2020 *)

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

A145988 Ending prime: n-th prime in the first occurrence of n consecutive primes of the form 4k + 3.

Original entry on oeis.org

3, 11, 223, 227, 491, 499, 503, 36607, 39703, 183283, 241727, 241739, 241771, 9177607, 9177611, 95949631, 105639463, 341118731, 727335359, 727335379, 1786054619, 1786054631, 22964264759, 54870713999, 79263248759

Offset: 1

Enoch Haga, Oct 26 2008

nonn

a(1)=3 is the same as A055624(1) because 3 is a single-digit number.

			a(2)=11 because this is the 2nd prime in the first run of 2 primes where p == 3 mod 4.

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

Cf. A055623, A054624, A145986, A145989, A145990, A145991, A145992, A145993, A145994.

Prime[#]&/@Flatten[Table[SequencePosition[If[Mod[#,4]==3,1,0]&/@Prime[ Range[ 615000]],PadRight[{},n,1],1],{n,15}],1][[All,2]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Jun 17 2022 *)

r=0; c=0; forprime(p=2, 4e9, if(p%4==3, if(c++>r, r=c; print1(p", ")), c=0)) \\ Charles R Greathouse IV, Mar 22 2011

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

Entry rewritten by, and a(14)-a(25) from, Charles R Greathouse IV, Mar 22 2011

A145990 Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).

Original entry on oeis.org

13, 37, 89, 109, 193, 229, 277, 313, 349, 389, 449, 509, 613, 661, 701, 757, 797, 853, 877, 929, 997, 1093, 1109, 1193, 1237, 1297, 1373, 1429, 1489, 1549, 1597, 1609, 1637, 1669, 1709, 1733, 1789, 1873, 1889, 1933, 1993, 2069, 2113, 2137, 2153, 2213, 2269

Offset: 1

Enoch Haga, Oct 26 2008

			a(1)=13 because this sequence includes consecutive runs of any length and this first term > 1 in a run of 2 is 13.

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. A039702, A055623, A054624, A145986, A145988 - A145994.

for i from 2 to 300 do
        if (ithprime(i) mod 4) = 1  and ithprime(i-1) mod 4 <> 1 and ithprime(i+1) mod 4 = 1 then
                printf("%d,",ithprime(i)) ;
        end if;
end do: # R. J. Mathar, Sep 30 2011

Prime[#+1]&/@(SequencePosition[Table[If[Mod[n,4]==1,1,0],{n,Prime[ Range[ 350]]}],{0,1,1},Overlaps->False][[All,1]]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)

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

Corrected and extended by Harvey P. Dale, Aug 02 2017

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

A145991 Final prime in a run of more than 1 consecutive primes == 1 (mod 4).

Original entry on oeis.org

17, 41, 101, 113, 197, 233, 281, 317, 353, 409, 461, 521, 617, 677, 709, 773, 809, 857, 881, 941, 1013, 1097, 1117, 1217, 1249, 1301, 1381, 1433, 1493, 1553, 1601, 1613, 1657, 1697, 1721, 1741, 1801, 1877, 1901, 1949, 1997, 2081, 2129, 2141, 2161, 2237

Offset: 1

Enoch Haga, Oct 26 2008

			a(1)=17 because this sequence includes consecutive runs of any length and this ending term > 1 in a run of 2 (comprising 13 and 17) is 17.

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

Cf. A055623, A054624, A145986, A145988- A145994

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

A145992 Run lengths of 2 or more consecutive primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 1

Enoch Haga, Oct 26 2008

			a(1) = 2 counts the two 3's from A039702(4) to A039702(5).
a(9) = 4 counts the four 3's from A039702(46) to A039702(49).
a(14)= 7 counts the seven 4's from A039702(90) to A039702(96).

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. A039702, A055623, A145986, A145988, A145989, A145990, A145991, A145993, A145994.

A145992 := proc()
    local m,p,r,i ;
    m := 3 ;
    p := 2 ;
    r := 0 ;
    for i from 2 to 1000 do
        if modp(p,4) = m then
            r := r+1 ;
        else
            if r > 1 then
                printf("%d,",r) ;
            end if;
            r := 0;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145992() ; # R. J. Mathar, Aug 29 2018

Most[Length /@ Select[ SplitBy[ Prime@ Range@ 780, Mod[#, 4] &], Mod[#[[1]], 4] == 3 && Length[#] > 1 &]] (* Giovanni Resta, Aug 29 2018 *)
Length/@Select[Split[Table[If[Mod[n,4]==3,1,0],{n,Prime[Range[ 1000]]}]], FreeQ[ #,0]&]/.(1->Nothing) (* Harvey P. Dale, Jul 27 2020 *)

Corrected by R. J. Mathar, Aug 29 2018

A145993 Primes that start a run of at least 2 consecutive primes of the form 4k+3.

Original entry on oeis.org

7, 19, 43, 67, 79, 103, 127, 163, 199, 307, 359, 379, 439, 463, 619, 643, 683, 719, 739, 823, 859, 883, 967, 983, 1087, 1163, 1279, 1303, 1423, 1439, 1459, 1483, 1499, 1559, 1663, 1783, 1811, 1867, 1979, 1999, 2083, 2099, 2179, 2239, 2347, 2399, 2447, 2531, 2579, 2659, 2683, 2699, 2803, 2843, 2879

Offset: 1

Enoch Haga, Oct 26 2008

			a(1)=7 because this sequence includes consecutive runs of any length and this first term >1 in a run of 2 is 7.

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

Cf. A039702, A055623, A145986, A145988, A145989, A145990, A145991, A145992 (run lengths) A145994.

A145993 := proc()
    local m,p,r,i,sp ;
    m := 3 ;
    p := 2 ;
    r := 0 ;
    sp := -1 ;
    for i from 2 to 1000 do
        if modp(p,4) = m then
            r := r+1 ;
            if r = 1 then
                sp := p ;
            end if;
        else
            if r > 1 then
                printf("%d,",sp) ;
            end if;
            r := 0;
            sp := -1 ;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145993() ; # R. J. Mathar, Aug 29 2018

Most[First /@ Select[ SplitBy[ Prime@ Range@ 425, Mod[#, 4] &], Mod[#[[1]], 4] == 3 && Length[#] > 1 &]] (* Giovanni Resta, Aug 29 2018 *)

619 inserted by R. J. Mathar, Aug 29 2018

A376396 Triangle read by rows: the n-th row gives the least sequence of n consecutive primes all of the form 4*m + 1.

Original entry on oeis.org

5, 13, 17, 89, 97, 101, 389, 397, 401, 409, 2593, 2609, 2617, 2621, 2633, 11593, 11597, 11617, 11621, 11633, 11657, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11689

Offset: 1

Stefano Spezia, Sep 22 2024

Guy writes that the terms of the 9th row have been found by De Haan. Moreover, Guy gives the terms of the 11th row: 766261, 766273, 766277, 766301, 766313, 766321, 766333, 766357, 766361, 766369, 766373.

			The triangle begins as:
       5;
      13,     17;
      89,     97,    101;
     389,    397,    401,    409;
    2593,   2609,   2617,   2621,   2633;
   11593,  11597,  11617,  11621,  11633,  11657;
   11593,  11597,  11617,  11621,  11633,  11657,  11677;
   11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681;
   11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681,  11689;
  373649, 373657, 373661, 373669, 373693, 373717, 373721, 373753, 373757, 373777;
  ...

R. K. Guy, Unsolved Problems in Number Theory, 2nd. ed., Section A4.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 11593 at. p. 173.

Jens Kruse Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373.

Cf. A002144, A016813, A057624 (1st column), A145986 (right hand column).

kold=1; row[n_]:=Module[{r={}}, k=kold; While[Mod[Prime[k],4]!=1, k++]; While[Product[Boole[Mod[Prime[k+i],4]==1], {i,0,n-1}]!=1, k++]; kold=k; Table[Prime[i+k], {i,0,n-1}]]; Array[row,9]//Flatten

cp's OEIS Frontend

A145994 Last prime in a run of at least 2 consecutive primes of the form 4k+3.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Mathematica

UBASIC

A145988 Ending prime: n-th prime in the first occurrence of n consecutive primes of the form 4k + 3.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

UBASIC

Extensions

A145990 Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

UBASIC

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

UBASIC

A145991 Final prime in a run of more than 1 consecutive primes == 1 (mod 4).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

UBASIC

A145992 Run lengths of 2 or more consecutive primes of the form 4k+3.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A145993 Primes that start a run of at least 2 consecutive primes of the form 4k+3.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple