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%I #9 Jun 29 2017 13:47:13
%S 1,3,2,6,4,7,8,12,5,11,9,14,16,15,13,19,21,22,10,23,17,24,18,27,29,30,
%T 26,35,52,38,25,31,32,28,33,47,34,46,20,39,40,44,36,43,37,48,41,75,53,
%U 59,61,54,57,55,58,60,64,51,65,56,66,62,50,71,45,78,42
%N Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.
%C A144795 gives the numbers without isolated 1's in base-2 representation.
%C This sequence is conjectured to be a permutation of the natural numbers.
%C This sequence has similarities with A269361: here we require that the product of two consecutive terms has no isolated 1, there the product of two consecutive terms has only isolated 1's, in base-2 representation.
%C For any k > 0:
%C - a(2*k-1) belongs to A091072,
%C - a(2*k) belongs to A091067.
%H Rémy Sigrist, <a href="/A289194/b289194.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A289194/a289194.gp.txt">PARI program for A289194</a>
%e The first terms, alongside a(n)*a(n+1) in binary, are:
%e n a(n) a(n)*a(n+1) in binary
%e -- ---- ---------------------
%e 1 1 11
%e 2 3 110
%e 3 2 1100
%e 4 6 11000
%e 5 4 11100
%e 6 7 111000
%e 7 8 1100000
%e 8 12 111100
%e 9 5 110111
%e 10 11 1100011
%e 11 9 1111110
%e 12 14 11100000
%e 13 16 11110000
%e 14 15 11000011
%e 15 13 11110111
%e 16 19 110001111
%e 17 21 111001110
%e 18 22 11011100
%e 19 10 11100110
%e 20 23 110000111
%t a = {1}; Do[k = 1; While[Nand[! MemberQ[a, k], ! MemberQ[Length /@ DeleteCases[Split[IntegerDigits[k Last[a], 2]], s_ /; First@ s == 0], 1]], k++]; AppendTo[a, k], {n, 2, 67}]; a (* _Michael De Vlieger_, Jun 29 2017 *)
%Y Cf. A091067, A091072, A144795, A269361.
%K nonn,base,look
%O 1,2
%A _Rémy Sigrist_, Jun 28 2017
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