A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.

5, 13, 89, 389, 2593, 11593, 11593, 11593, 11593, 373649, 766261, 3358169, 12204889, 12270077, 12270077, 12270077, 297387757, 297779117, 297779117, 1113443017, 1113443017, 1113443017, 1113443017, 1113443017, 84676452781, 84676452781, 689101181569, 689101181569, 689101181569, 3278744415797, 3278744415797, 3278744415797, 3278744415797

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.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 163.

Links

Table of n, a(n) for n=1..33.

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(9) = 11593 because "[t]his number is the first in a sequence of 9 consecutive primes all of the form 4n + 1."

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 ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 19} ]

Crossrefs

Cf. A057619, A057620, A057622, A055623, A055624.

Sequence in context: A263468 A350467 A081560 * A092567 A055623 A280294

Adjacent sequences: A057621 A057622 A057623 * A057625 A057626 A057627

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