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%I #33 Oct 19 2017 03:13:52
%S 5,23,47,251,1889,7793,43451,243161,726893,759821,1820111,1820111,
%T 10141499,19725473,19725473,136209239,400414121,400414121,489144599,
%U 489144599,766319189,766319189,21549657539,21549657539,21549657539,140432294381,140432294381,437339303279,1871100711071,3258583681877
%N Initial prime in first sequence of n consecutive primes congruent to 5 modulo 6.
%C 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
%C The sequence is infinite, by Shiu's theorem. - _Jonathan Sondow_, Jun 22 2017
%D R. K. Guy, "Unsolved Problems in Number Theory", A4
%H Giovanni Resta, <a href="/A057622/b057622.txt">Table of n, a(n) for n = 1..35</a> (terms < 4*10^14)
%H J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H D. K. L. Shiu, <a href="http://dx.doi.org/10.1112/S0024610799007863">Strings of Congruent Primes</a>, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [<a href="http://www.ams.org/mathscinet-getitem?mr=1760689">MR1760689</a>]
%F a(n) = A000040(A247967(n)). a(n) = min { A055626(k); k >= n }. - _M. F. Hasler_, Sep 03 2016
%e a(12) = 1820111 because this number is the first in a sequence of 12 consecutive primes all of the form 6n + 5.
%t 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 *)
%t 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 *)
%Y Cf. A057619, A057620, A057624.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Oct 09 2000
%E More terms from _Don Reble_, Nov 16 2003
%E More terms from _Jens Kruse Andersen_, May 30 2006
%E Three lines of data (derived from J.K.Andersen's web page) completed by _M. F. Hasler_, Sep 02 2016
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