%I #15 Apr 29 2024 09:30:02

%S 1,2,4,6,8,10,3,9,12,14,5,15,18,21,24,27,16,20,7,28,25,30,33,36,22,26,

%T 35,42,32,34,39,45,38,40,44,46,49,56,48,50,55,60,51,54,11,66,52,58,62,

%U 64,13,65,63,70,57,69,68,72,17,85,75,78,80,90,19,76,74

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

%C This sequence is a permutation of the positive integers with inverse A372185; the proof is similar to that for A370843.

%H Rémy Sigrist, <a href="/A372144/b372144.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A372144/a372144.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+1), a(2*n+1), a(2*n+2)) <> 1 for any n > 0.

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

%e |

%e 1

%e |

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

%e | |

%e .---4---. .---6---.

%e | | | |

%e .-8-. .10-. .-3-. .-9-.

%e | | | | | | | |

%e 12 14 5 15 18 21 24 27

%o (PARI) \\ See Links section.

%Y See A370843 and A372030 for similar sequences.

%Y Cf. A372130 (analog without common prime factor), A372143 (analog based on binary 1's), A372185 (inverse).

%K nonn

%O 1,2

%A _Rémy Sigrist_, Apr 20 2024