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%I #11 Jun 08 2012 00:48:08
%S 1,2,3,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,24,25,26,27,29,
%T 30,31,33,34,35,37,38,39,40,41,42,43,45,46,47,48,49,50,51,53,54,55,56,
%U 57,58,59,61,62,63,64,65,66,67,69,70,71,72,73,74,75,77,78,79,80,81,82,83,85,86,87,88,89,90,91,93,94,95,97,98,99,101,102,103,104,105
%N a(1) = 1, a(2) = 2 and, for n > 2, a(n) is the smallest integer greater than a(n - 1) such that no three terms of the sequence form a geometric progression of the form {x, 2 x, 4 x}.
%C Conjecture. The positive integers that are not in this sequence are given by the positions of 2 in the fixed-point of the morphism 0->01, 1->02, 2->03, 3->01 (see A191255). (This has been confirmed for over 5000 terms of A213257.) To illustrate, the fixed-point of the indicated morphism is {0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,1,0,1,0,2,0,...} and 2 occurs at positions {4,12,20,...}, integers that are missing in A213257.
%C The positive integers that are not in this sequence are listed in A213258.
%C For the sequence containing no 3-term arithmetic progression,see A003278.
%e Given that the sequence begins {1, 2, 3, 5, 6, 7, 8, 9, 10, 11,...}, the next term, a(11), cannot be 12, because then the forbidden progression {3,6,12} would occur in the sequence. 13 is allowed, however, so a(11)=13.
%Y Cf. A003278, A191255, A213258.
%K nonn
%O 1,2
%A _John W. Layman_, Jun 07 2012