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Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, the binary expansions of a(n), a(2*n) and a(2*n+1) have a common 1 bit.
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%I #13 Apr 29 2024 09:29:54

%S 0,1,3,5,2,6,4,7,10,11,12,13,14,15,9,17,8,24,18,19,20,21,22,23,25,26,

%T 27,28,29,30,16,31,40,41,42,43,34,35,33,37,36,38,39,44,45,46,47,49,48,

%U 50,51,52,53,54,55,56,57,58,59,60,61,62,63,65,32,96,67,69

%N Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, the binary expansions of a(n), a(2*n) and a(2*n+1) have a common 1 bit.

%C Conjecture: this sequence is a permutation of the nonnegative integers.

%H Rémy Sigrist, <a href="/A372143/b372143.txt">Table of n, a(n) for n = 0..9999</a>

%H Rémy Sigrist, <a href="/A372143/a372143.gp.txt">PARI program</a>

%F a(n) AND a(2*n) AND a(2*n+1) <> 0 for any n > 0 (where AND denotes the bitwise AND operator).

%e The first terms, arranged alongside a binary tree where each parent node (except the root) and its children share some 1 bit, are:

%e |

%e 0

%e |

%e .-------1-------.

%e | |

%e .---3---. .---5---.

%e | | | |

%e .-2-. .-6-. .-4-. .-7-.

%e | | | | | | | |

%e 10 11 12 13 14 15 9 17

%o (PARI) \\ See Links section.

%Y See A372030 for similar sequences.

%Y Cf. A372129 (analog without common 1 bit).

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Apr 20 2024