cp's OEIS Frontend

A085515 Order of first occurrence of a sequence of exactly n consecutive primes of the form 6*k+1.

%I A085515 #24 Jul 19 2025 14:14:18
%S A085515 1,2,3,6,4,5,7,8,10,11,9,13,12,15,16,14,17,18,21,19,20,23,22,25,24,26,
%T A085515 31,27,28,30,29,32,34,33,35
%N A085515 Order of first occurrence of a sequence of exactly n consecutive primes of the form 6*k+1.
%C A085515 This sequence gives the index of A055625(n) after sorting by increasing magnitude.
%H A085515 J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H A085515 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a085515.txt">Record producing runs of primes of the forms 6k+-1</a>.
%H A085515 Hugo Pfoertner, <a href="http://www.enginemonitoring.net/math/isprim.txt">FORTRAN function ISPRIM to check primality</a>.
%e A085515 a(3)=3, a(4)=6, a(5)=4 because the first occurrence of exactly 6 consecutive primes of the form 6*k+1 (1741,1747,1753,1759,1777,1783) occurs after the first occurrence of 3 consecutive primes of this form (151,157,163) and before the first occurrence of 4 consecutive "6k+1"-primes (3049,3061,3067,3079).
%t A085515 (* Program not suitable to compute a large number of terms. *)
%t A085515 pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^7}];
%t A085515 sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
%t A085515 b[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 1} ..}]& ][[1, 1]];
%t A085515 Sort[Table[{b[n], n}, {n, 1, 16}]][[All, 2]] (* _Jean-François Alcover_, Nov 21 2018 *)
%o A085515 (Fortran) ! Program given at link.
%Y A085515 Cf. A055625, A085516 (sorted occurrence of first runs of "6k-1" primes).
%K A085515 more,nonn
%O A085515 1,2
%A A085515 _Hugo Pfoertner_, Jul 31 2003
%E A085515 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006
%E A085515 a(32)-a(35) from _Giovanni Resta_, Aug 04 2013