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

1,1

Comments

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

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) = A000040(A247967(n)). a(n) = min { A055626(k); k >= n }. - M. F. Hasler, Sep 03 2016

Example

a(12) = 1820111 because this number is the first in a sequence of 12 consecutive primes all of the form 6n + 5.

Mathematica

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 *)

Crossrefs

Cf. A057619, A057620, A057624.

Sequence in context: A154625 A107011 A031387 * A055626 A127200 A147113

Adjacent sequences: A057619 A057620 A057621 * A057623 A057624 A057625

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

Three lines of data (derived from J.K.Andersen's web page) completed by M. F. Hasler, Sep 02 2016

Status

approved