A317024 Lexicographically earliest sequence of positive terms such that for any distinct i and j, lcm(a(i), a(i+1)) and lcm(a(j), a(j+1)) are distinct.

1, 1, 2, 3, 1, 4, 3, 5, 1, 7, 2, 5, 4, 7, 3, 8, 1, 9, 2, 11, 1, 13, 2, 15, 4, 9, 5, 7, 6, 11, 3, 13, 4, 11, 5, 8, 7, 9, 8, 11, 7, 10, 9, 11, 10, 13, 5, 16, 1, 17, 2, 19, 1, 23, 2, 25, 1, 27, 2, 29, 1, 31, 2, 32, 3, 16, 7, 12, 11, 13, 6, 17, 3, 19, 4, 17, 5, 19

Offset: 1

Author

Rémy Sigrist, Jul 19 2018

Keywords

• nonn

Comments

See A317025 for the corresponding LCM.

This sequence has similarities with A088177.

For any n > 0, let g(n) = gcd(a(n), a(n+1)); between 1 and 800000, the function g takes only 5 times a value other than 1.

For any n > 0 and prime number p, if p divides a(n+1), then the p-adic valuation of a(n+1) is strictly greater than the p-adic valuation of a(n).

This sequence contains infinitely many distinct values.

The first occurrence of a prime number p, if not preceded by 1, is followed by 1.

The first occurrence of a prime power k, if not preceded by a divisor of k, is followed by 1.

If this sequence contains infinitely many 1's, then A317025 is a permutation of the natural numbers.

Examples

			The first terms, alongside lcm(a(n), a(n+1)), are:
  n  a(n)  lcm(a(n), a(n+1))
  -- ----  -----------------
   1    1    1
   2    1    2
   3    2    6
   4    3    3
   5    1    4
   6    4   12
   7    3   15
   8    5    5
   9    1    7
  10    7   14
  11    2   10
  12    5   20
  13    4   28
  14    7   21
  15    3   24
  16    8    8
  17    1    9
  18    9   18
  19    2   22
  20   11   11
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A317024
• Rémy Sigrist, Scatterplot of the ordinal transform of the first 500000 terms

Crossrefs

Cf. A088177, A317025.

Programs

PARI

See Links section.