A372144 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.

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, 35, 42, 32, 34, 39, 45, 38, 40, 44, 46, 49, 56, 48, 50, 55, 60, 51, 54, 11, 66, 52, 58, 62, 64, 13, 65, 63, 70, 57, 69, 68, 72, 17, 85, 75, 78, 80, 90, 19, 76, 74

Offset

1,2

Comments

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

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Formula

GCD(a(n+1), a(2*n+1), a(2*n+2)) <> 1 for any n > 0.

Example

The first terms, arranged alongside a binary tree where each parent node (except the root) and its children share some prime factor, are:
                 |
                 1
                 |
         .-------2-------.
         |               |
     .---4---.       .---6---.
     |       |       |       |
   .-8-.   .10-.   .-3-.   .-9-.
   |   |   |   |   |   |   |   |
  12  14   5  15  18  21  24  27

Prog

(PARI) \\ See Links section.

Crossrefs

See A370843 and A372030 for similar sequences.

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

Sequence in context: A004522 A004521 A103694 * A357379 A083167 A338741

Adjacent sequences: A372141 A372142 A372143 * A372145 A372146 A372147

Keyword

nonn

Author

Rémy Sigrist, Apr 20 2024

Status

approved