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A284726 a(n) = (1/4) * smallest multiple of 4 missing from [A280864(1), ..., A280864(n-1)].

%I A284726 #19 Nov 26 2017 09:51:28
%S A284726 1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,7,7,8,8,8,8,8,8,8,8,8,9,9,9,10,
%T A284726 10,10,10,10,10,10,10,11,11,11,11,11,14,14,14,14,14,14,14,14,14,14,14,
%U A284726 15,15,15,15,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,20,20,20,21,21,21,22,22
%N A284726 a(n) = (1/4) * smallest multiple of 4 missing from [A280864(1), ..., A280864(n-1)].
%C A284726 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_4(n)/4.
%C A284726 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 A284726 Rémy Sigrist, <a href="/A284726/b284726.txt">Table of n, a(n) for n = 1..10000</a>
%H A284726 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 A284726 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 A284726 The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing multiple of 3 in [1,2,4,3,6] is 8, so a(6) = 8/4 = 2.
%p A284726 mex := proc(L)
%p A284726 local k;
%p A284726 for k from 1 do
%p A284726 if not k in L then
%p A284726 return k;
%p A284726 end if;
%p A284726 end do:
%p A284726 end proc:
%p A284726 read b280864;
%p A284726 k:=4; a:=[1,1]; ML:=[]; B:=1;
%p A284726 for n from 2 to 120 do
%p A284726 t:=b280864[n];
%p A284726 if (t mod k) = 0 then
%p A284726 ML:=[op(ML),t/k];
%p A284726 B:=mex(ML);
%p A284726 a:=[op(a),B];
%p A284726 else
%p A284726 a:=[op(a),B];
%p A284726 fi;
%p A284726 od:
%p A284726 a;
%t A284726 terms = 84; rad[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A284726 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 A284726 Clear[a]; a[1] = 1;
%t A284726 a[n_] := a[n] = For[b = 4 a[n - 1], True, b += 4, If[FreeQ[A280864[[1 ;; n - 1]], b], Return[b/4]]];
%t A284726 Array[a, terms] (* _Jean-François Alcover_, Nov 26 2017, after _Rémy Sigrist_'s program for A280864 *)
%Y A284726 Cf. A280864, A064413, A284724, A280725.
%K A284726 nonn
%O A284726 1,4
%A A284726 _N. J. A. Sloane_, Apr 06 2017