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.

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, 26, 35, 52, 38, 25, 31, 32, 28, 33, 47, 34, 46, 20, 39, 40, 44, 36, 43, 37, 48, 41, 75, 53, 59, 61, 54, 57, 55, 58, 60, 64, 51, 65, 56, 66, 62, 50, 71, 45, 78, 42

Offset: 1

Author

Rémy Sigrist, Jun 28 2017

Keywords

• nonn
• base
• look

Comments

A144795 gives the numbers without isolated 1's in base-2 representation.

This sequence is conjectured to be a permutation of the natural numbers.

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.

For any k > 0:

- a(2*k-1) belongs to A091072,

- a(2*k) belongs to A091067.

Examples

			The first terms, alongside a(n)*a(n+1) in binary, are:
n       a(n)    a(n)*a(n+1) in binary
--      ----    ---------------------
1       1       11
2       3       110
3       2       1100
4       6       11000
5       4       11100
6       7       111000
7       8       1100000
8       12      111100
9       5       110111
10      11      1100011
11      9       1111110
12      14      11100000
13      16      11110000
14      15      11000011
15      13      11110111
16      19      110001111
17      21      111001110
18      22      11011100
19      10      11100110
20      23      110000111
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A289194

Crossrefs

Cf. A091067, A091072, A144795, A269361.

Programs

Mathematica

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 *)