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

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

1,1

Comments

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

References

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

Links

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]

Formula

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

Example

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A057619, A057622, A057624, A055625.

Sequence in context: A003526 A121517 A199216 * A055625 A102239 A188233

Adjacent sequences: A057617 A057618 A057619 * A057621 A057622 A057623

Keyword

nonn

Author

Robert G. Wilson v, Oct 09 2000

Extensions

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

Status

approved