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A284724 a(n) = (1/2) * smallest even number missing from [A280864(1), ..., A280864(n-1)].

%I A284724 #27 Nov 23 2017 19:42:19
%S A284724 1,1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,10,12,13,13,13,15,15,15,15,16,16,
%T A284724 16,16,18,18,18,20,20,20,20,20,20,20,20,22,22,22,22,22,25,25,25,25,25,
%U A284724 27,27,27,28,28,28,30,30,30,30,32,32,32,32,33,33,33,35,35,35,35,36,36,36,36,40,40,40,42
%N A284724 a(n) = (1/2) * smallest even number missing from [A280864(1), ..., A280864(n-1)].
%C A284724 For k>=1, n>=1, let B_k(n) = smallest multiple of k missing from [A280864(1), ..., A280864(n-1)]. Sequence gives values of B_2(n)/2.
%C A284724 The analogous sequences B_k(n) for the EKG sequence A064413 were important for the analysis of that sequence, so they may also be useful for studying A280864.
%H A284724 Rémy Sigrist, <a href="/A284724/b284724.txt">Table of n, a(n) for n = 1..10000</a>
%H A284724 J. C. Lagarias, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0204011">The EKG sequence</a>, arXiv:math/0204011 [math.NT], 2002.
%H A284724 J. C. Lagarias, E. M. Rains and N. J. A. Sloane, <a href="http://www.emis.de/journals/EM/expmath/volumes/11/11.3/Lagarias437_446.pdf">The EKG Sequence</a>, Exper. Math. 11 (2002), 437-446.
%e A284724 The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing even number from [1,2,4,3,6] is 8, so a(6) = 8/2 = 4.
%p A284724 mex := proc(L)
%p A284724         local k;
%p A284724         for k from 1 do
%p A284724                 if not k in L then
%p A284724                         return k;
%p A284724                 end if;
%p A284724         end do:
%p A284724 end proc:
%p A284724 read b280864;
%p A284724 k:=2; a:=[1,1]; ML:=[]; B:=1;
%p A284724 for n from 2 to 120 do
%p A284724 t:=b280864[n];
%p A284724 if (t mod k) = 0 then
%p A284724 ML:=[op(ML),t/k];
%p A284724 B:=mex(ML);
%p A284724 a:=[op(a),B];
%p A284724 else
%p A284724 a:=[op(a),B];
%p A284724 fi;
%p A284724 od:
%p A284724 a;
%t A284724 terms = 80; rad[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A284724 A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]];
%t A284724 Clear[a];
%t A284724 a[1] = 1;
%t A284724 a[n_] := a[n] = For[b = 2a[n-1], True, b += 2, If[FreeQ[A280864[[1 ;; n-1]], b], Return[b/2]]];
%t A284724 Array[a, terms] (* _Jean-François Alcover_, Nov 23 2017, after _Rémy Sigrist_ program for A280864 *)
%Y A284724 Cf. A280864, A064413, A284725, A284726.
%K A284724 nonn
%O A284724 1,3
%A A284724 _N. J. A. Sloane_, Apr 06 2017