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A057619 Initial prime in first sequence of n primes congruent to 3 modulo 4. 6
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 (list; graph; refs; listen; history; text; internal format)
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)
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
Sequence in context: A105763 A291352 A132564 * A349006 A349838 A179859
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

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Last modified March 28 13:17 EDT 2024. Contains 371254 sequences. (Running on oeis4.)