A057622 -id:A057622 - cp's OEIS frontend

A098058 Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).

2, 3, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 367, 373, 383, 409, 419, 421, 431

Offset: 1

Cino Hilliard, Sep 11 2004

First differences are also not divisible by 4. - Zak Seidov, Jun 23 2015
Starting with 3, group the primes into runs of consecutive primes either all == 1 (mod 4) or all == 3 (mod 4). Only the last prime of each run appears in this sequence. Since the runs alternate == 1 (mod 4) and == 3 (mod 4), so do the members of this sequence. - Franklin T. Adams-Watters, Jun 23 2015
The sequence is infinite, by Dirichlet's theorem on primes in arithmetic progressions. The sequence contains arbitrarily long gaps, by Daniel Shiu's theorem on strings of congruent primes (see A057619 and A057622). Conjecture: The sequence contains arbitrarily long strings of consecutive primes (see A289118). - Jonathan Sondow, Jun 25 2017

			Prime(2) = 3, prime(3) = 5. 4 does not divide 5-3 so prime(2)=3 is in the sequence.
Runs: (3), (5), (7,11), (17), (19, 23), (29), (31), (37,41), (43,47), (53), ... The sequence is 2 followed by the last member of each run. Differences within each run are always divisible by 4.

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Consecutive Congruent Primes

Cf. A098059, A289118.

Prime[Select[Range[100], Mod[Prime[ # + 1] - Prime[ # ], 4] !=0 &]] (* Ray Chandler, Oct 09 2006 *)

f(n) = for(x=1,n,z=(prime(x+1)-prime(x));if(z%4,print1(prime(x)",")))

alist(n)=my(r=vector(n),p=2,np,k=0);while(kFranklin T. Adams-Watters, Jun 23 2015

list(lim)=my(v=List(),p=2); forprime(q=3,nextprime(lim\1+1), if((q-p)%4, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jun 24 2015

Edited by Ray Chandler, Oct 26 2006

A057620 Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.

Original entry on oeis.org

7, 31, 151, 1741, 1741, 1741, 19471, 118801, 148531, 148531, 406951, 2339041, 2339041, 51662593, 51662593, 73451737, 232301497, 450988159, 1444257673, 1444257673, 1444257673, 24061965043, 24061965043, 43553959717, 43553959717

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

See A055626 for the variant "exactly n". See A247967 for the indices of these primes. See A057620, A057621 for variants "congruent to 5 (mod 6)", resp. "(mod 3)". - M. F. Hasler, Sep 03 2016

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

			a(6) = 1741 because this number is the first in a sequence of 6 consecutive primes all of the form 3n + 1.

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, A057622, A057624, A055625.

p = 0; Do[a = Table[-1, {n}]; k = Max[1, p]; While[Union[a] != {1}, k = NextPrime[k]; a = Take[AppendTo[a, Mod[k, 3]], -n]]; p = NestList[NextPrime[#, -1] &, k, n]; Print[p[[-2]]]; p = p[[-1]], {n, 1, 18}] (* Robert G. Wilson v, updated by Michael De Vlieger, Sep 03 2016 *)
Table[k = 1; While[Total@ Boole@ Map[Mod[#, 6] == 1 &, NestList[NextPrime, Prime@ k, n - 1]] != n, k++]; Prime@ k, {n, 12}] (* Michael De Vlieger, Sep 03 2016 *)

m=c=o=0; forprime(p=1,, p%6 != 1 && (!c||!c=0) && next; c||o=p; c++>m||next; m++; print1(", ",o)) \\ M. F. Hasler, Sep 03 2016

a(n) <= A055625(n). - Zak Seidov, Aug 29 2016

a(n) = A000040(A247967(n)). a(n) = min { A055625(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
Definition clarified by Zak Seidov, Jun 19 2017

A247967 a(n) is the smallest k such that prime(k+i) (mod 6) takes successively the values 5, 5, ... for i = 0, 1, ..., n-1.

Original entry on oeis.org

3, 9, 15, 54, 290, 987, 4530, 21481, 58554, 60967, 136456, 136456, 673393, 1254203, 1254203, 7709873, 21357253, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 68745739764, 117327812949

Offset: 1

Michel Lagneau, Sep 28 2014

nonn

Weakening the definition to prime(k+i) == 2 (mod 3) yields a(1) = 1, but all other terms are unchanged. See also A247816 (residue 5) or A276414 (equal residues, all 1 or all -1). - M. F. Hasler, Sep 02 2016

			a(1)= 3 => prime(3) == 5 (mod 6).
a(2)= 9 => prime(9) == 5 (mod 6), prime(10) == 5 (mod 6).
a(3)= 15 => prime(15) == 5 (mod 6), prime(16) == 5 (mod 6), prime(17) == 5 (mod 6).
From _Michel Marcus_, Sep 30 2014: (Start)
The resulting primes are:
  5;
  23, 29;
  47, 53, 59;
  251, 257, 263, 269;
  1889, 1901, 1907, 1913, 1931;
  7793, 7817, 7823, 7829, 7841, 7853;
  43451, 43457, 43481, 43487, 43499, 43517, 43541;
  243161, 243167, 243197, 243203, 243209, 243227, 243233, 243239;
  ... (End)

Amiram Eldar, Table of n, a(n) for n = 1..35

Cf. A057622, A247816, A276414.

N = 2*10^8; % to use primes up to N
P = mod(primes(N),6);
P5 = find(P==5);
n5 = numel(P5);
a(1) = P5(1);
for k = 2:100
  r = find(P5(k:n5) == P5(1:n5+1-k)+k-1,1,'first');
  if numel(r) == 0
     break
  end
  a(k) = P5(r);
end
a % Robert Israel, Sep 02 2016

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 = 5
        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:

Table[k = 1; While[Times @@ Boole@ Map[Mod[Prime[k + #], 6] == 5 &, Range[0, n - 1]] == 0, k++]; k, {n, 10}] (* Michael De Vlieger, Sep 02 2016 *)

a(n) = {k = 1; ok = 0; while (!ok, nb = 0; for (i=0, n-1, if (prime(k+i) % 6 == 5, nb++, break);); if (nb == n, ok=1, k++);); k;} \\ Michel Marcus, Sep 28 2014

m=c=i=0;forprime(p=1,, i++;p%6!=5&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

a(n) = primepi(A057622(n)). - Michel Marcus, Oct 01 2014

a(11)-a(22) from A057622 by Michel Marcus, Oct 03 2014
a(23)-a(25) from Jinyuan Wang, Jul 08 2019
a(26)-a(30) added using A057622 by Jinyuan Wang, Apr 15 2020

A055626 First prime starting a chain of exactly n consecutive primes congruent to 5 modulo 6.

Original entry on oeis.org

5, 23, 47, 251, 1889, 7793, 43451, 243161, 726893, 759821, 2280857, 1820111, 10141499, 40727657, 19725473, 136209239, 744771077, 400414121, 1057859471, 489144599, 13160911739, 766319189, 38451670931, 119618704427, 21549657539, 141116164769, 140432294381, 437339303279

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 5 modulo 6.
a(21)>2^31, a(22)= 766319189. - Hugo Pfoertner, Jul 31 2003
See A057622 for the variant where "exactly" is replaced by "at least". See A055625 for the variant "congruent to 1 (mod 6)". - 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.

Cf. A055623, A055624, A055625, A085516.

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, 5} ..}]& ][[1, 1]];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 13}] (* Jean-François Alcover, Nov 21 2018 *)

okchain(n, p) = {if ((precprime(p-1) % 6) == 5, return (0)); for (i=1, n, if ((p % 6) != 5, return (0)); p = nextprime(p+1);); if ((p % 6) == 5, 0, 1);}
a(n) = {p = 5; while (! okchain(n, p), p = nextprime(p+1)); p;} \\ Michel Marcus, Dec 17 2013

a(9)-a(13), including correction of a(9)-a(10) from Reiner Martin, Jul 18 2001
a(14)-a(20) from Hugo Pfoertner, Jul 31 2003
a(21)-a(25) from Jens Kruse Andersen, May 30 2006
a(26) and beyond from Giovanni Resta, Aug 04 2013

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

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

A057621 Initial prime in first sequence of n primes congruent to 2 modulo 3.

Original entry on oeis.org

2, 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

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

Same as A057622 except for a(1). - Jens Kruse Andersen, May 30 2006

			a(12) = 1820111 because this number is the first in a sequence of 12 consecutive primes all of the form 3*n + 2.

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.

Cf. A057622, A000040, A247967.

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 ] != {2}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 3 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 18} ]

a(n) = A000040(A247967(n)) for all n > 1. - M. F. Hasler, Sep 03 2016

More terms from Jens Kruse Andersen, May 30 2006

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

A284101 Initial primes of 14 consecutive primes all congruent to 5 mod 6.

Original entry on oeis.org

19725473, 19725479, 40727657, 40962983, 40962989, 44291297, 45404537, 45404543, 54470123, 54470147, 63846089, 63846119, 68208599, 68208611, 68364221, 90007661, 93602507, 99492971, 99492977, 99579647, 99579671, 117585977, 117585983

Offset: 1

Zak Seidov, Mar 20 2017

nonn

Prime indices of a(n): 1254203,1254204,2475221,2488622,2488623,2677942,2741244,2741245.

a(1) = 19725473 = A057622(14). Actually 19725473, 40962983 and 45404537 are the initial primes of 15 such consecutive primes that is also a(1) = A057622(15).

			15 consecutive primes starting with 19725473 are
{19725473, 19725479, 19725527, 19725533, 19725599, 19725617, 19725623, 19725653, 19725659, 19725677, 19725683, 19725689, 19725701, 19725731, 19725737} with gaps
{6, 48, 6, 66, 18, 6, 30, 6, 18, 6, 6, 12, 30, 6} - all multiples of 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A057622 Initial prime in first sequence of n consecutive primes all congruent to 5 mod 6.

Select[Partition[Prime[Range[68*10^5]],14,1],AllTrue[Mod[#,6]==5&]][[;;,1]] (* Harvey P. Dale, Mar 06 2023 *)
Prime[#]&/@(SequencePosition[If[Mod[#,6]==5,1,0]&/@Prime[Range[672*10^4]],PadRight[{},14,1]][[;;,1]]) (* Harvey P. Dale, Sep 22 2024 *)

list(lim)=my(v=List(), P=primes(14), goal=vector(14,i,5), pm=P%6, idx=1); forprime(p=P[#P]+1,, P[idx]=p; pm[idx]=p%6; if(idx++>14,idx=1); if(P[idx]>lim, break); if(pm==goal, listput(v, P[idx]))); Vec(v) \\ Charles R Greathouse IV, Mar 20 2017

a(9)-a(23) from Charles R Greathouse IV, Mar 20 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

cp's OEIS Frontend

A098058 Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Extensions

A057620 Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A247967 a(n) is the smallest k such that prime(k+i) (mod 6) takes successively the values 5, 5, ... for i = 0, 1, ..., n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

PARI

Formula

Extensions

A055626 First prime starting a chain of exactly n consecutive primes congruent to 5 modulo 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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.