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A176466 The smallest prime q which stays prime through at least 3 iterations of q -> 2 * q + prime(n+1).

%I A176466 #6 May 21 2025 12:59:44
%S A176466 2,13,5,199,2,13,251,487,61,5,113,19,2,13,157,1621,269,23,139,557,5,
%T A176466 37,241,5,19,587,823,41,97,5,109,13,1151,31,1409,53,5,1543,67,421,5,
%U A176466 1039,2,13,41,359,1697,43,101,157,1531,179,79,193,37,181,149,113,4519,197,397,23,739,2,283,29,5,163,1031,1987
%N A176466 The smallest prime q which stays prime through at least 3 iterations of q -> 2 * q + prime(n+1).
%C A176466 See comments and references of A176379.
%C A176466 q, 2 * q + prime(n+1), 4 * q + 3 * prime(n+1) and 8 * q + 7 * prime(n+1) are required to be prime.
%C A176466 List of (q,first iteration, 2nd iteration, 3rd iteration):
%C A176466 (2,7,17,37) (13,31,67,139) (5,17,41,89) (199,409,829,1669) (2,17,47,107)
%C A176466 (13,43,103,223) (251,521,1061,2141) (487,997,2017,4057) (61,151,331,691) (5,41,113,257)
%C A176466 (113,263,563,1163) (19,79,199,439) (2,47,137,317) (13,73,193,433) (157,367,787,1627)
%C A176466 (1621,3301,6661,13381) (269,599,1259,2579) (23,113,293,653) (139,349,769,1609) (557,1187,2447,4967)
%C A176466 (5,89,257,593) (37,157,397,877) (241,571,1231,2551) (5,107,311,719) (19,139,379,859)
%C A176466 (587,1277,2657,5417) (823,1753,3613,7333) (41,191,491,1091) (97,307,727,1567) (5,137,401,929)
%C A176466 (109,349,829,1789) (13,163,463,1063) (1151,2441,5021,10181) (31,211,571,1291) (1409,2969,6089,12329)
%C A176466 (53,263,683,1523) (5,173,509,1181) (1543,3253,6673,13513) (67,307,787,1747) (421,1021,2221,4621)
%C A176466 (5,191,563,1307) (1039,2269,4729,9649) (2,197,587,1367) (13,223,643,1483) (41,281,761,1721)
%C A176466 (359,929,2069,4349) (1697,3617,7457,15137) (43,313,853,1933) (101,431,1091,2411) (157,547,1327,2887)
%e A176466 n=1: q=2, iteration 2 * q + prime(2) = 7, iteration 2 * 7 + 3 = 17, 2 * 17 + 3 = 37: q=2 is first term
%e A176466 n=2: q=13, iteration 2 * 13 + prime(3) = 31, iteration 2 * 31 + 5 = 67, iteration 2 * 67 + 5 = 139, q=13 is 2nd term
%p A176466 A176466 := proc(n)
%p A176466     pk1 := ithprime(n+1) ;
%p A176466     for pidx from 1 do
%p A176466         p := ithprime(pidx) ;
%p A176466         pitr := 2*p+pk1 ;
%p A176466         if not isprime(pitr) then
%p A176466             next ;
%p A176466         end if;
%p A176466         pitr := 2*pitr+pk1 ;
%p A176466         if not isprime(pitr) then
%p A176466             next ;
%p A176466         end if;
%p A176466         pitr := 2*pitr+pk1 ;
%p A176466         if not isprime(pitr) then
%p A176466             next ;
%p A176466         else
%p A176466             return p ;
%p A176466         end if;
%p A176466     end do:
%p A176466 end proc:
%p A176466 seq(A176466(n),n=1..80) ; # _R. J. Mathar_, May 21 2025
%Y A176466 Cf. A005602, A059761, A059762, A110024 , A176379
%K A176466 base,nonn
%O A176466 1,1
%A A176466 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 18 2010