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

1,2

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.

Links

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

Rémy Sigrist, PARI program for A338619

Example

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

Prog

(PARI) See Links section.

Crossrefs

See A338618 for a similar sequence.

Cf. A084937, A098550.

Sequence in context: A332214 A285322 A129593 * A279355 A279356 A241909

Adjacent sequences: A338616 A338617 A338618 * A338620 A338621 A338622

Keyword

nonn

Author

Rémy Sigrist, Nov 04 2020

Status

approved