A057621 -id:A057621 - 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

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

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

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