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A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

%I A289194 #9 Jun 29 2017 13:47:13
%S A289194 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 A289194 26,35,52,38,25,31,32,28,33,47,34,46,20,39,40,44,36,43,37,48,41,75,53,
%U A289194 59,61,54,57,55,58,60,64,51,65,56,66,62,50,71,45,78,42
%N A289194 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 A289194 A144795 gives the numbers without isolated 1's in base-2 representation.
%C A289194 This sequence is conjectured to be a permutation of the natural numbers.
%C A289194 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 A289194 For any k > 0:
%C A289194 - a(2*k-1) belongs to A091072,
%C A289194 - a(2*k) belongs to A091067.
%H A289194 Rémy Sigrist, <a href="/A289194/b289194.txt">Table of n, a(n) for n = 1..10000</a>
%H A289194 Rémy Sigrist, <a href="/A289194/a289194.gp.txt">PARI program for A289194</a>
%e A289194 The first terms, alongside a(n)*a(n+1) in binary, are:
%e A289194 n       a(n)    a(n)*a(n+1) in binary
%e A289194 --      ----    ---------------------
%e A289194 1       1       11
%e A289194 2       3       110
%e A289194 3       2       1100
%e A289194 4       6       11000
%e A289194 5       4       11100
%e A289194 6       7       111000
%e A289194 7       8       1100000
%e A289194 8       12      111100
%e A289194 9       5       110111
%e A289194 10      11      1100011
%e A289194 11      9       1111110
%e A289194 12      14      11100000
%e A289194 13      16      11110000
%e A289194 14      15      11000011
%e A289194 15      13      11110111
%e A289194 16      19      110001111
%e A289194 17      21      111001110
%e A289194 18      22      11011100
%e A289194 19      10      11100110
%e A289194 20      23      110000111
%t A289194 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 A289194 Cf. A091067, A091072, A144795, A269361.
%K A289194 nonn,base,look
%O A289194 1,2
%A A289194 _Rémy Sigrist_, Jun 28 2017