A360613 Lexicographically earliest sequence of positive integers such that the products of the form a(2*u-1) * a(2*v) with u, v > 0 are all distinct.

1, 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 14, 17, 18, 19, 23, 24, 25, 29, 26, 31, 28, 33, 36, 37, 41, 40, 43, 47, 46, 49, 50, 51, 52, 53, 59, 55, 61, 57, 63, 64, 67, 71, 73, 79, 81, 83, 82, 85, 86, 87, 88, 89, 91, 93, 92, 95, 97, 101, 100, 103, 107, 109, 113, 111

Offset: 1

Author

Rémy Sigrist, Feb 14 2023

Keywords

• nonn

Comments

In other words, the products of a term from the odd bisection by a term from the even bisection are all distinct.

If we consider the bitwise XOR operator instead of the multiplication then we obtain A000695 interleaved with A062880.

The value 1 is the only duplicate.

All prime numbers appear in this sequence, in ascending order.

For n = 1..50000, if m_n denotes the least positive value not in {a(2*u-1) * a(2*v), 1 <= 2*u-1 <= n and 1 <= 2*v <= n}, then a(n+1) = m_n or a(n+2) = m_n. Will this pattern last forever?

Examples

			The first terms, alongside the corresponding products, are:
  n   a(n)  Corresponding products
  --  ----  --------------------------
   1     1
   2     1   1
   3     2   2
   4     3   3,  6
   5     4   4, 12
   6     5   5, 10, 20
   7     7   7, 21, 35
   8     8   8, 16, 32,  56
   9     9   9, 27, 45,  72
  10    11  11, 22, 44,  77,  99
  11    13  13, 39, 65, 104, 143
  12    15  15, 30, 60, 105, 135, 195
		

Links

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

Crossrefs

Cf. A000695, A062880, A066724, A360627-A360628 (bisections), A360633 (products).

Programs

C

See Links section.

Formula

a(n) < a(n+2).