A338619 Lexicographically earliest sequence of distinct positive terms such that among three consecutive terms there is exactly one pair of terms that are not coprime.

1, 2, 4, 3, 8, 9, 10, 5, 7, 14, 11, 6, 12, 13, 15, 18, 17, 16, 20, 19, 22, 24, 23, 21, 27, 25, 33, 35, 28, 29, 26, 30, 31, 32, 34, 37, 36, 38, 41, 40, 42, 43, 39, 45, 44, 46, 47, 48, 50, 49, 52, 54, 53, 51, 57, 55, 63, 56, 59, 58, 60, 61, 62, 64, 65, 66, 68

Offset: 1

Author

Rémy Sigrist, Nov 04 2020

Keywords

• nonn

Comments

In other words, for any n > 0, exactly 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                 4                   1
   4     3                 1                 3                   1
   5     8                 1                 2                   1
   6     9                 1                 1                   5
   7    10                 5                 1                   1
   8     5                 1                 1                   7
   9     7                 7                 1                   1
  10    14                 1                 2                   1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A338619

Crossrefs

See A338618 for a similar sequence.

Cf. A084937, A098550.

Programs

PARI

See Links section.