A057619 Initial prime in first sequence of n primes congruent to 3 modulo 4.

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

1,1

Comments

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..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]

Example

a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.

Mathematica

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 *)

Crossrefs

Cf. A057620, A057622, A057624.

Sequence in context: A105763 A291352 A132564 * A349006 A349838 A179859

Adjacent sequences: A057616 A057617 A057618 * A057620 A057621 A057622

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 29 2006

Status

approved