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A251416 a(n) = smallest number not in {A098550(1), A098550(2), ..., A098550(n)}.

%I A251416 #35 Aug 01 2018 09:21:31
%S A251416 2,3,4,5,5,5,5,5,6,7,7,7,7,7,10,11,11,11,11,11,11,13,17,17,17,17,17,
%T A251416 17,17,18,19,19,19,19,19,19,19,19,19,19,19,19,23,23,23,23,23,23,23,23,
%U A251416 29,29,29,29,29,29,29,29,29,29,31
%N A251416 a(n) = smallest number not in {A098550(1), A098550(2), ..., A098550(n)}.
%C A251416 a(n) = smallest missing number in A098550 once we have found A098550(n).
%C A251416 _Bradley Klee_ conjectures that after a(30)=18, all further terms are primes, that every prime appears, and the primes appear in increasing order.
%D A251416 Bradley Klee, Posting to Sequence Fans Mailing List, Dec 03 2014
%H A251416 N. J. A. Sloane and Reinhard Zumkeller, <a href="/A251416/b251416.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from N. J. A. Sloane)
%F A251416 a(n) = Min{A251546(n), A251549(n)}. - _Reinhard Zumkeller_, Dec 19 2014
%p A251416 # This produces the first 100 terms. Uses b1 = list of terms in A098550, from b-file
%p A251416 b2:={$3..5000}:
%p A251416 b3:=[2]:
%p A251416 for i from 2 to 100 do
%p A251416 b2:=remove('x->x=b1[i]',b2):
%p A251416 b3:=[op(b3),b2[1]];
%p A251416 od:
%p A251416 b3;
%t A251416 terms = 100;
%t A251416 f[lst_List] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
%t A251416 A098550 = Nest[f, {1, 2, 3}, terms-3];
%t A251416 a[1] = 2;
%t A251416 a[n_] := a[n] = For[k = a[n - 1], True, k++, If[FreeQ[A098550[[1 ;; n]], k], Return[k]]];
%t A251416 Array[a, terms] (* _Jean-François Alcover_, Aug 01 2018, after _Robert G. Wilson v_ *)
%o A251416 (Haskell)
%o A251416 import Data.List (delete)
%o A251416 a251416 n = a251416_list !! (n-1)
%o A251416 a251416_list = 2 : 3 : f 2 3 [4..] where
%o A251416    f u v ws = h ws where
%o A251416      h (x:xs) = if gcd x u > 1 && gcd x v == 1
%o A251416                    then (head ws) : f v x (delete x ws) else h xs
%o A251416 -- _Reinhard Zumkeller_, Dec 05 2014
%Y A251416 Cf. A098550, A251415. See A251417 for lengths of runs.
%Y A251416 Cf. A251595 (distinct terms).
%Y A251416 Cf. A251546, A251549.
%K A251416 nonn
%O A251416 1,1
%A A251416 _N. J. A. Sloane_, Dec 03 2014