A320123 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

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

Rémy Sigrist, PARI program for A320123

EXAMPLE

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