A057624 -id:A057624 - 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

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

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

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 *)

A057632 Initial prime in first sequence of n primes congruent to 3 modulo 8.

Original entry on oeis.org

3, 491, 2243, 42299, 274123, 4310083, 4310083, 9065867, 547580443, 1885434347, 8674616939, 11312238283, 19201563659, 619849118491, 4056100954547, 13721202685691, 119254168189363, 276151474703651, 2189798979924331, 3153425741761723

Offset: 1

Robert G. Wilson v, Oct 10 2000

			a(3) = 2243 because this number is the first in a sequence of 3 consecutive primes all of the form 8*n + 3.

J. K. Andersen, Consecutive Congruent Primes.

Cf. A363017 (indices), A057624 (with 1 modulo 4).

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, 8 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 9} ]

More terms from Jens Kruse Andersen, May 28 2006
a(16)-a(18) from Giovanni Resta, Aug 04 2013
a(19)-a(20) from Martin Ehrenstein, May 28 2023

A363016 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 1 mod 4.

Original entry on oeis.org

3, 6, 24, 77, 378, 1395, 1395, 1395, 1395, 31798, 61457, 240748, 800583, 804584, 804584, 804584, 16118548, 16138563, 16138563, 56307979, 56307979, 56307979, 56307979, 56307979, 3511121443, 3511121443, 26284355567, 26284355567, 26284355567, 118027458557, 118027458557, 118027458557, 118027458557

Offset: 1

Léo Gratien, May 13 2023

nonn

a(n) is also the minimal rank where n consecutive 1's appear in A023512.

The sequence is infinite by Shiu's theorem.

			For n=3, a(3) = 24 because prime(24)+1=90, prime(25)+1=98, and prime(26)+1=102 are the first 3 consecutive primes p such that p+1 is divisible by 2 and not by 4.

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

Cf. A000720, A057624, A023512.

Cf. A363017 (3 mod 8).

a(n) = A000720(A057624(n)). - Amiram Eldar, May 13 2023

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

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

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A057620 Initial prime in first sequence of 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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A162865 Initial prime of exactly nine consecutive primes congruent to 1 modulo 4.

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A057632 Initial prime in first sequence of n primes congruent to 3 modulo 8.

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A363016 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 1 mod 4.

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

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