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

Author

Robert G. Wilson v, Oct 09 2000

Keywords

• nonn

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

Examples

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

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]

Crossrefs

Cf. A057619, A057622, A057624, A055625.

Programs

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 *)

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

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

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