This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A057620 #35 Oct 19 2017 03:13:52
%S A057620 7,31,151,1741,1741,1741,19471,118801,148531,148531,406951,2339041,
%T A057620 2339041,51662593,51662593,73451737,232301497,450988159,1444257673,
%U A057620 1444257673,1444257673,24061965043,24061965043,43553959717,43553959717
%N A057620 Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.
%C A057620 See A055626 for the variant "exactly n". See A247967 for the indices of these primes. See A057620, A057621 for variants "congruent to 5 (mod 6)", resp. "(mod 3)". - _M. F. Hasler_, Sep 03 2016
%C A057620 The sequence is infinite, by Shiu's theorem. - _Jonathan Sondow_, Jun 22 2017
%D A057620 R. K. Guy, "Unsolved Problems in Number Theory", A4
%H A057620 Giovanni Resta, <a href="/A057620/b057620.txt">Table of n, a(n) for n = 1..35</a> (terms < 4*10^14)
%H A057620 J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H A057620 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 A057620 a(n) <= A055625(n). - _Zak Seidov_, Aug 29 2016
%F A057620 a(n) = A000040(A247967(n)). a(n) = min { A055625(k); k >= n }. - _M. F. Hasler_, Sep 03 2016
%e A057620 a(6) = 1741 because this number is the first in a sequence of 6 consecutive primes all of the form 3n + 1.
%t A057620 p = 0; Do[a = Table[-1, {n}]; k = Max[1, p]; While[Union[a] != {1}, k = NextPrime[k]; a = Take[AppendTo[a, Mod[k, 3]], -n]]; p = NestList[NextPrime[#, -1] &, k, n]; Print[p[[-2]]]; p = p[[-1]], {n, 1, 18}] (* _Robert G. Wilson v_, updated by _Michael De Vlieger_, Sep 03 2016 *)
%t A057620 Table[k = 1; While[Total@ Boole@ Map[Mod[#, 6] == 1 &, NestList[NextPrime, Prime@ k, n - 1]] != n, k++]; Prime@ k, {n, 12}] (* _Michael De Vlieger_, Sep 03 2016 *)
%o A057620 (PARI) m=c=o=0; forprime(p=1,, p%6 != 1 && (!c||!c=0) && next; c||o=p; c++>m||next; m++; print1(", ",o)) \\ _M. F. Hasler_, Sep 03 2016
%Y A057620 Cf. A057619, A057622, A057624, A055625.
%K A057620 nonn
%O A057620 1,1
%A A057620 _Robert G. Wilson v_, Oct 09 2000
%E A057620 More terms from _Don Reble_, Nov 16 2003
%E A057620 More terms from _Jens Kruse Andersen_, May 30 2006
%E A057620 Definition clarified by _Zak Seidov_, Jun 19 2017