A057620 -id:A057620 - cp's OEIS frontend

A057622 Initial prime in first sequence of n consecutive primes congruent to 5 modulo 6.

5, 23, 47, 251, 1889, 7793, 43451, 243161, 726893, 759821, 1820111, 1820111, 10141499, 19725473, 19725473, 136209239, 400414121, 400414121, 489144599, 489144599, 766319189, 766319189, 21549657539, 21549657539, 21549657539, 140432294381, 140432294381, 437339303279, 1871100711071, 3258583681877

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

Same as A057621 except for a(1). See A057620 for primes congruent to 1 (mod 6). See A055626 for the variant "exactly n", which is an upper bound, cf. formula. - M. F. Hasler, Sep 03 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

			a(12) = 1820111 because this number is the first in a sequence of 12 consecutive primes all of the form 6n + 5.

R. K. Guy, "Unsolved Problems in Number Theory", A4

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A057619, A057620, A057624.

p = 0; Do[a = Table[-1, {n}]; k = Max[1, p]; While[Union@ a != {5}, k = NextPrime@ k; a = Take[AppendTo[a, Mod[k, 6]], -n]]; p = NestList[NextPrime[#, -1] &, k, n]; Print[p[[-2]]]; p = p[[-1]], {n, 18}] (* Robert G. Wilson v, updated by Michael De Vlieger, Sep 03 2016 *)
Table[k = 1; While[Total@ Boole@ Map[Mod[#, 6] == 5 &, NestList[NextPrime, Prime@ k, n - 1]] != n, k++]; Prime@ k, {n, 12}] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = A000040(A247967(n)). a(n) = min { A055626(k); k >= n }. - M. F. Hasler, Sep 03 2016

More terms from Don Reble, Nov 16 2003
More terms from Jens Kruse Andersen, May 30 2006
Three lines of data (derived from J.K.Andersen's web page) completed by M. F. Hasler, Sep 02 2016

A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.

Original entry on oeis.org

5, 13, 89, 389, 2593, 11593, 11593, 11593, 11593, 373649, 766261, 3358169, 12204889, 12270077, 12270077, 12270077, 297387757, 297779117, 297779117, 1113443017, 1113443017, 1113443017, 1113443017, 1113443017, 84676452781, 84676452781, 689101181569, 689101181569, 689101181569, 3278744415797, 3278744415797, 3278744415797, 3278744415797

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

			a(9) = 11593 because "[t]his number is the first in a sequence of 9 consecutive primes all of the form 4n + 1."

R. K. Guy, Unsolved Problems in Number Theory, A4.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 163.

Jens Kruse Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A057619, A057620, A057622, A055623, A055624.

NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 19} ]

More terms from Don Reble, Nov 16 2003

More terms from Jens Kruse Andersen, May 29 2006

A055625 First prime starting a chain of exactly n consecutive primes congruent to 1 modulo 6.

Original entry on oeis.org

7, 31, 151, 3049, 7351, 1741, 19471, 118801, 498259, 148531, 406951, 2513803, 2339041, 89089369, 51662593, 73451737, 232301497, 450988159, 1558562197, 2506152301, 1444257673, 28265029657, 24061965043, 87996684091, 43553959717

Offset: 1

Labos Elemer, Jun 05 2000

nonn

The term "exactly" means that before the first and after the last primes of chain, the immediate primes are not congruent to 1 modulo 6.

See A057620 for the variant where "exactly" is replaced by "at least". - M. F. Hasler, Sep 03 2016

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

Cf. A055623, A055624, A055626, A085515.

Fortran
```
c See link in A085515.
```

pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^6}];
sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
a[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 1} ..}]& ][[1, 1]];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 13}] (* Jean-François Alcover, Nov 21 2018 *)

Corrected and extended by Reiner Martin, May 19 2001
More terms from Hugo Pfoertner, Jul 31 2003
a(20)>2^31, a(21)=1444257673. - Hugo Pfoertner, Jul 31 2003
More terms from Jens Kruse Andersen, May 30 2006
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

A057619 Initial prime in first sequence of n primes congruent to 3 modulo 4.

Original entry on oeis.org

3, 7, 199, 199, 463, 463, 463, 36551, 39607, 183091, 241603, 241603, 241603, 9177431, 9177431, 95949311, 105639091, 341118307, 727334879, 727334879, 1786054147, 1786054147, 22964264027, 54870713243, 79263248027, 113391385603

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

			a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.

R. K. Guy, "Unsolved Problems in Number Theory", A4

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A057620, A057622, A057624.

NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {3}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 18} ]
With[{prs=Table[If[Mod[Prime[n],4]==3,1,0],{n,4646*10^6}]},Prime/@ Table[ SequencePosition[prs,PadRight[{},k,1],1][[1,1]],{k,26}]] (* The program will take a long time to run and requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 28 2017 *)

More terms from Don Reble, Nov 16 2003

More terms from Jens Kruse Andersen, May 29 2006

A054679 First of n consecutive primes which differ by a multiple of 6.

Original entry on oeis.org

2, 23, 47, 251, 1741, 1741, 19471, 118801, 148531, 148531, 406951, 1820111, 2339041, 19725473, 19725473, 73451737, 232301497, 400414121, 489144599, 489144599, 766319189, 766319189, 21549657539, 21549657539, 21549657539, 140432294381, 140432294381, 437339303279, 1552841185921, 1552841185921, 1552841185921

Offset: 1

Jeff Burch, Apr 18 2000

nonn

See A276414 for the indices of these primes. - M. F. Hasler, Sep 02 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14) (corrected by Ray Chandler, Jan 19 2019)
J. K. Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A054678, A054680; A276414.

a(n) = A000040(A276414(n)). - M. F. Hasler, Sep 02 2016

a(n) = min(A057620(n), A057621(n)) for all n >= 1. - M. F. Hasler, Sep 03 2016

More terms from Larry Reeves (larryr(AT)acm.org), Nov 09 2000
More terms from Jens Kruse Andersen, May 30 2006
Initial term and a(27)-a(31) added and name edited by M. F. Hasler, Sep 02 2016

A247816 a(n) is the smallest k such that prime(k+i) = 1 (mod 6) for i = 0, 1,...,n-1.

Original entry on oeis.org

4, 11, 36, 271, 271, 271, 2209, 11199, 13717, 13717, 34369, 172146, 172146, 3094795, 3094795, 4308948, 12762142, 23902561, 72084956, 72084956, 72084956, 1052779161, 1052779161, 1857276773, 1857276773, 19398320447, 57446769091, 57446769091, 57446769091

Offset: 1

Michel Lagneau, Sep 28 2014

Equivalently, "mod 6" can be replaced by "mod 3". See A247967 for the variant "= 5 (mod 6)" and A276414 for runs of primes congruent to each other (mod 3). - M. F. Hasler, Sep 02 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

			a(1)= 4 => prime(4) (mod 6)= 1;
a(2)= 11 => prime(11)(mod 6)= 1, prime(12)(mod 6) = 1;
a(3)= 36 => prime(36)(mod 6)= 1, prime(37)(mod 6)= 1, prime(38)(mod 6)= 1.
The resulting primes are:
7;
31, 37;
151, 157, 163;
1741, 1747, 1753, 1759;
1741, 1747, 1753, 1759, 1777;
1741, 1747, 1753, 1759, 1777, 1783;
19471, 19477, 19483, 19489, 19501, 19507, 19531;
... - _Michel Marcus_, Sep 29 2014

Giovanni Resta, Table of n, a(n) for n = 1..35
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A057620, A247967; A276414.

for n from 1 to 22 do :
ii:=0:
   for k from 3 to 10^5 while (ii=0)do :
     s:=0:
      for i from 0 to n-1 do:
        r:=irem(ithprime(k+i),6):
        if r = 1
        then
        s:=s+1:
        else
        fi:
      od:
       if s=n and ii=0
       then
       printf ( "%d %d \n",n,k):ii:=1:
       else
       fi:
    od:
od:

With[{m6=If[Mod[#,6]==1,1,0]&/@Prime[Range[5*10^6]]},Flatten[Table[SequencePosition[ m6,PadRight[{},n,1],1],{n,16}],1]][[;;,1]] (* Harvey P. Dale, May 07 2023 *)

m=c=i=0;forprime(p=1,, i++;p%6!=1&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

a(n) = primepi(A057620(n)). - Michel Marcus, Sep 30 2014

a(12)-a(21) from A057620 by Michel Marcus, Oct 03 2014

a(22)-a(29) from Giovanni Resta, Oct 03 2018

A289119 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6k+1 and 6k+5 or 6k+5 and 6k+1.

Original entry on oeis.org

5, 5, 5, 5, 5, 5, 5, 89, 89, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 3954889, 15186319, 15186319, 15186319, 77011289, 77011289, 77011289, 288413159, 288413159, 288413159, 288413159, 288413159, 62585146739, 114058236679, 143014298809

Offset: 1

Jonathan Sondow, Jun 25 2017

nonn

Conjecture: the sequence is infinite. (Motivation: the string HTHTHT... of length n eventually occurs in any sufficiently long sequence of coin tosses.)

			For k = 3, 4, ..., 33, {Prime[k], Mod[Prime[k], 6]} = {5, 5}, {7, 1}, {11, 5}, {13, 1}, {17, 5}, {19, 1}, {23, 5}, {29, 5}, {31, 1}, {37, 1}, {41, 5}, {43, 1}, {47, 5}, {53, 5}, {59, 5}, {61, 1}, {67,  1}, {71, 5}, {73, 1}, {79, 1}, {83, 5}, {89, 5}, {97, 1}, {101, 5}, {103, 1}, {107, 5}, {109, 1}, {113, 5}, {127, 1}, {131, 5}, {137, 5}, so a(n) = 5, 5, 5, 5, 5, 5, 5, 89, 89 for n = 1, 2, ..., 9 with a(10) > 89.

R. K. Guy, Unsolved Problems in Number Theory, A4.

Giovanni Resta, Table of n, a(n) for n = 1..43
Jens Kruse Andersen, Consecutive Congruent Primes

Cf. A057620, A057622, A289118.

j = 3; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] == 0, j++]; Prime[j], {n, 0, 20}]; Prepend[T, 5]

a(23)-a(36) from Giovanni Resta, Jun 29 2017

A288915 Run lengths in A039704.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Jun 19 2017

nonn

Is the sequence bounded?

On Dickson's conjecture this sequence is unbounded. Records: a(1) = 1, a(9) = 2, a(13) = 3, a(39) = 4, a(180) = 6, a(1348) = 7, a(6698) = 8, a(8156) = 10, a(20230) = 11, a(79011) = 12, a(99250) = 13, a(710895) = 15, a(2421600) = 16, a(7128444) = 17, a(11898707) = 18, a(14368535) = 20, a(21943755) = 22, a(519775979) = 25, a(3111006505) = 27. - Charles R Greathouse IV, Jun 19 2017

Zak Seidov, Table of n, a(n) for n = 1..20000

Cf. A039704, A057620, A057622.

Length /@ Split[Mod[Prime[Range[100]], 6]]

t=1;p=2;forprime(q=3,1e3,if((q-p)%6==0,t++,print1(t", ");t=1);p=q) \\ Charles R Greathouse IV, Jun 19 2017

a(70) corrected by Charles R Greathouse IV, Jun 19 2017

A382249 a(n) is the smallest starting prime of a sequence of exactly n consecutive primes that are alternately of the form 6k+1 and 6k-1 or vice versa.

Original entry on oeis.org

23, 19, 17, 13, 11, 7, 5, 97, 89, 877, 863, 859, 857, 853, 839, 829, 827, 823, 821, 811, 809, 3954889, 15186331, 15186323, 15186319, 77011331, 77011303, 77011289, 288413249, 288413233, 288413219, 288413173, 288413159, 62585146739, 114058236679, 143014298851, 143014298831, 143014298809

Offset: 1

Jean-Marc Rebert, Mar 19 2025

nonn

			a(1) = 23, because 23 and 29 are 2 consecutive primes such that 23 = 6*4 - 1, while 29 = 6*5 - 1. Additionally, no smaller prime possesses this property.
a(2) = 19, because 19, 23 and 29 are 3 consecutive primes such that 19 = 6*3 + 1 and 23 = 6*4 - 1, while 29 = 6*5 - 1. Additionally, no smaller prime possesses this property.
Table of consecutive primes
  1 [23] = 6*[4] + [-1];
  2 [19, 23] = 6*[3, 4] + [1, -1];
  3 [17, 19, 23] = 6*[3, 3, 4] + [-1, 1, -1];
  4 [13, 17, 19, 23] = 6*[2, 3, 3, 4] + [1, -1, 1, -1];
  5 [11, 13, 17, 19, 23] = 6*[2, 2, 3, 3, 4] + [-1, 1, -1, 1, -1];

Cf. A057620, A057622.

cp's OEIS Frontend

A057622 Initial prime in first sequence of n consecutive primes congruent to 5 modulo 6.

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A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.

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A055625 First prime starting a chain of exactly n consecutive primes congruent to 1 modulo 6.

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A057619 Initial prime in first sequence of n primes congruent to 3 modulo 4.

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A054679 First of n consecutive primes which differ by a multiple of 6.

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A247816 a(n) is the smallest k such that prime(k+i) = 1 (mod 6) for i = 0, 1,...,n-1.

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A289119 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1.

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A289119 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6k+1 and 6k+5 or 6k+5 and 6k+1.

A382249 a(n) is the smallest starting prime of a sequence of exactly n consecutive primes that are alternately of the form 6k+1 and 6k-1 or vice versa.