A320123 a(1) = 1, a(2) = 2, a(3) = 3, and for any n > 3, a(n) = the smallest positive integer not yet in the sequence such that gcd(a(n-2), a(n-1)), gcd(a(n-1), a(n)) and gcd(a(n), a(n-2)) are all distinct.

1, 2, 3, 6, 4, 9, 12, 8, 15, 10, 14, 5, 20, 16, 25, 30, 18, 27, 22, 24, 11, 33, 21, 7, 36, 28, 35, 26, 40, 13, 52, 32, 39, 42, 34, 17, 38, 68, 19, 76, 44, 55, 45, 48, 46, 23, 50, 92, 60, 51, 56, 54, 49, 63, 57, 70, 66, 65, 75, 69, 80, 72, 78, 64, 81, 84, 58

Offset: 1

Author

Rémy Sigrist, Oct 06 2018

Keywords

• nonn

Comments

This sequence is a variant of A127202.

The sequence is well defined as for any n > 3, provided the first n terms are known, any number v of the form a(n-2) * b where b is coprime to a(n) * a(n-1) satisfies #{ gcd(a(n-1), a(n)), gcd(a(n), v), gcd(v, a(n-1)) } = #{ gcd(a(n-1), a(n)), gcd(a(n), a(n-2)), gcd(a(n-2), a(n-1)) } = 3, and a(n+1) exists.

In the scatterplot of the sequence, the prime numbers correspond to the lower line.

Examples

			The first terms, alongside gcd(a(n-2), a(n-1)), gcd(a(n-1), a(n)) and gcd(a(n), a(n-2)), are:
  n   a(n)  gcd(a(n-2),a(n-1))  gcd(a(n-1),a(n))  gcd(a(n),a(n-2))
  --  ----  ------------------  ----------------  ----------------
   1     1                 N/A               N/A               N/A
   2     2                 N/A                 1               N/A
   3     3                   1                 1                 1
   4     6                   1                 3                 2
   5     4                   3                 2                 1
   6     9                   2                 1                 3
   7    12                   1                 3                 4
   8     8                   3                 4                 1
   9    15                   4                 1                 3
  10    10                   1                 5                 2
  11    14                   5                 2                 1
  12     5                   2                 1                 5
  13    20                   1                 5                 2
  14    16                   5                 4                 1
  15    25                   4                 1                 5
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A320123

Crossrefs

Cf. A127202.

Programs

PARI

See Links section.