A338618 Lexicographically earliest sequence of distinct positive integers such that three consecutive terms are never pairwise coprime.

1, 2, 4, 3, 6, 5, 8, 10, 7, 12, 9, 11, 15, 18, 13, 14, 16, 17, 20, 22, 19, 24, 21, 23, 27, 30, 25, 26, 28, 29, 32, 34, 31, 36, 33, 35, 39, 40, 38, 37, 42, 44, 41, 46, 48, 43, 45, 50, 47, 52, 54, 49, 51, 56, 57, 58, 60, 53, 55, 65, 59, 70, 62, 61, 64, 66, 63

Offset: 1

Author

Rémy Sigrist, Nov 04 2020

Keywords

• nonn

Comments

In other words, for any n > 0, at least one of gcd(a(n), a(n+1)), gcd(a(n), a(n+2)), gcd(a(n+1), a(n+2)) is strictly greater than 1.

This sequence has connections with the Yellowstone permutation (A098550).

Conjecture: this sequence is a permutation of the natural numbers.

Examples

			The first terms, alongside associated GCD's, are:
  n   a(n)  gcd(a(n),a(n+1))  gcd(a(n),a(n+2))  gcd(a(n+1),a(n+2))
  --  ----  ----------------  ----------------  ------------------
   1     1                 1                 1                   2
   2     2                 2                 1                   1
   3     4                 1                 2                   3
   4     3                 3                 1                   1
   5     6                 1                 2                   1
   6     5                 1                 5                   2
   7     8                 2                 1                   1
   8    10                 1                 2                   1
   9     7                 1                 1                   3
  10    12                 3                 1                   1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A338618

Crossrefs

See A338619 for a similar sequence.

Cf. A084937, A098550.

Programs

PARI

See Links section.