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

1,3

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.

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

Example

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

Prog

(PARI) See Links section.

Crossrefs

Cf. A088177, A317025.

Sequence in context: A161621 A095701 A067992 * A354803 A140757 A258254

Adjacent sequences: A317021 A317022 A317023 * A317025 A317026 A317027

Keyword

nonn

Author

Rémy Sigrist, Jul 19 2018

Status

approved