%I #14 Apr 29 2024 09:29:41

%S 1,2,3,5,7,4,11,6,13,8,9,15,17,10,19,23,25,12,29,21,31,14,37,16,41,18,

%T 35,27,43,20,33,22,39,24,47,49,53,26,45,32,55,28,51,57,59,30,61,63,65,

%U 34,67,71,73,36,79,38,77,40,69,81,83,46,85,75,89,44,95

%N Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(n), a(2*n) and a(2*n+1) have distinct prime factors.

%C This sequence is a permutation of the positive integers with inverse A372132:

%C - for any prime number p, the first multiple of p is precisely p,

%C - all prime numbers appear in the sequence, in increasing order,

%C - for any v > 0, each prime number not dividing v provides an opportunity to select v later, and eventually v will appear in the sequence.

%H Rémy Sigrist, <a href="/A372130/b372130.txt">Table of n, a(n) for n = 1..10001</a>

%H Rémy Sigrist, <a href="/A372130/a372130.gp.txt">PARI program</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F GCD(a(n), a(2*n)) = GCD(a(n), a(2*n+1)) = GCD(a(2*n), a(2*n+1)) = 1 for any n > 0.

%e The first terms, arranged alongside a binary tree where siblings have distinct prime factors and parent and children have distinct prime factors, are:

%e |

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

%e | |

%e .---2---. .---3---.

%e | | | |

%e .-5-. .-7-. .-4-. .11-.

%e | | | | | | | |

%e 6 13 8 9 15 17 10 19

%o (PARI) \\ See Links section.

%Y See A372030 for similar sequences.

%Y Cf. A372129 (analog based on binary 1's), A372132 (inverse).

%K nonn

%O 1,2

%A _Rémy Sigrist_, Apr 20 2024