A372130 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.

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, 35, 27, 43, 20, 33, 22, 39, 24, 47, 49, 53, 26, 45, 32, 55, 28, 51, 57, 59, 30, 61, 63, 65, 34, 67, 71, 73, 36, 79, 38, 77, 40, 69, 81, 83, 46, 85, 75, 89, 44, 95

Offset

1,2

Comments

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

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

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

- 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.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10001

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Formula

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.

Example

The first terms, arranged alongside a binary tree where siblings have distinct prime factors and parent and children have distinct prime factors, are:
                |
        .-------1-------.
        |               |
    .---2---.       .---3---.
    |       |       |       |
  .-5-.   .-7-.   .-4-.   .11-.
  |   |   |   |   |   |   |   |
  6  13   8   9  15  17  10  19

Prog

(PARI) \\ See Links section.

Crossrefs

See A372030 for similar sequences.

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

Sequence in context: A373999 A114319 A125151 * A302024 A273665 A212646

Adjacent sequences: A372127 A372128 A372129 * A372131 A372132 A372133

Keyword

nonn

Author

Rémy Sigrist, Apr 20 2024

Status

approved