A055623 -id:A055623 - cp's OEIS frontend

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

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

A162865 Initial prime of exactly nine consecutive primes congruent to 1 modulo 4.

Original entry on oeis.org

11593, 206953, 315257, 541097, 906541, 992393, 1124993, 1410361, 1595081, 1781569, 1872049, 2043329, 2090353, 2312749, 2381657, 2481509, 2497289, 2718389, 2758109, 2772409, 2976397, 3863473, 3868849, 4027957, 4042673, 4375141, 4464841, 4547581, 4606153

Offset: 1

Rick L. Shepherd, Jul 15 2009

The table provides all 8919 [=A092660(9)] terms less than 10^9.

If 10 or more consecutive primes are all congruent to 1 modulo 4, none of them is a member of this sequence. - Harvey P. Dale, Oct 23 2011

Rick L. Shepherd, Table of n, a(n) for n=1..8919

Cf. A092660, A162866, A055623, A057624, A054678.

m9Q[l_]:=Module[{ms=Mod[l,4]},First[ms]!=1&&Last[ms]!=1&&Union[Take[ ms,{2,10}]]=={1}]; Transpose[Select[Partition[ Prime[Range[ 290000]], 11,1],m9Q]][[2]] (* Harvey P. Dale, Oct 23 2011 *)

A330360 First occurrence of run of lucky numbers congruent to 1 mod 4 of exactly length n.

Original entry on oeis.org

1, 9, 285, 933, 741, 11173, 15109, 33705, 100737, 24025, 34197, 86829, 5370693, 6457761, 2287341, 26529033, 53099457, 770289, 754577025, 256655365, 765951429, 2618761237, 1075872265, 2405972445, 2767592133

Offset: 1

Amiram Eldar, Dec 12 2019

Calculated using Hugo van der Sanden's Lucky numbers up to 10^9 (private communication).
a(20) = 256655365.
a(26) > 4*10^9. - Giovanni Resta, May 10 2020

			a(1) = 1 since 1 is the first lucky number congruent to 1 mod 4, and the next lucky number is 3 which is not congruent to 1 mod 4.
a(2) = 9 since 9 and 13 are 2 consecutive lucky numbers congruent to 1 mod 4, following 7 and followed by 15 which are both not congruent to 1 mod 4.

Amiram Eldar and Giovanni Resta, Table of n, k, A000959(k),... A000959(k+n-1) for n=1..25

Cf. A000959, A055623, A137168, A330359, A330361.

a(19)-a(25) from Giovanni Resta, May 10 2020

A035525 24 consecutive primes of form 4k+1.

Original entry on oeis.org

1113443017, 1113443029, 1113443053, 1113443069, 1113443077, 1113443137, 1113443141, 1113443153, 1113443173, 1113443189, 1113443209, 1113443249, 1113443269, 1113443293, 1113443329, 1113443341, 1113443369, 1113443381, 1113443389, 1113443473, 1113443521, 1113443533, 1113443549, 1113443561

Offset: 1

Carlos Rivera

This is the earliest sequence of this form (L=24).

See Section A4. p. 13 "Unsolved problems in Number Theory", 2nd. Ed., R. K. Guy

C. K. Caldwell, Prime Curios! 1113443017
Carlos Rivera, Puzzle 256. Jack Brennen old records, The Prime Puzzles and Problems Connection. Longer sequences.
Carlos Rivera, Puzzle 539. n consecutive primes same residue modulo 4, The Prime Puzzles and Problems Connection. Longer sequences.
Hakan Summakoğlu, Puzzle 619. Odd midpoints, The Prime Puzzles and Problems Connection. Mentions this sequence.

Cf. A035518, A035498, A055623.

More terms from Sean A. Irvine, Oct 15 2020

A226905 First in a sequence of 9 consecutive primes all of the form 4n+1.

Original entry on oeis.org

11593, 206953, 315257, 373649, 373657, 495377, 495389, 509389, 509393, 541097, 612109, 612113, 766261, 766273, 766277, 789097, 789101, 906541, 992393, 1124993, 1330229, 1330237, 1410361, 1531633, 1531657, 1531661, 1578133, 1578169, 1595081, 1694897, 1694909

Offset: 1

Harvey P. Dale, Jun 21 2013

nonn

			206953, 206993, 207013, 207017, 207029, 207037, 207041, 207061, and 207073 are 9 consecutive primes, each equal to 1 mod 4.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 163 (entry for 11593).

Cf. A055623 (first occurrence of run of primes congruent to 1 mod 4 of exactly length n).

Transpose[Select[Partition[Prime[Range[180000]],9,1],Union[Mod[#,4]] == {1}&]][[1]]

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

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

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

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Maple

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A162865 Initial prime of exactly nine consecutive primes congruent to 1 modulo 4.

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Mathematica

A330360 First occurrence of run of lucky numbers congruent to 1 mod 4 of exactly length n.

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A035525 24 consecutive primes of form 4k+1.

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A226905 First in a sequence of 9 consecutive primes all of the form 4n+1.

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Mathematica