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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.

%I A360613 #19 Feb 25 2023 15:20:01
%S A360613 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,
%T A360613 41,40,43,47,46,49,50,51,52,53,59,55,61,57,63,64,67,71,73,79,81,83,82,
%U A360613 85,86,87,88,89,91,93,92,95,97,101,100,103,107,109,113,111
%N 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.
%C A360613 In other words, the products of a term from the odd bisection by a term from the even bisection are all distinct.
%C A360613 If we consider the bitwise XOR operator instead of the multiplication then we obtain A000695 interleaved with A062880.
%C A360613 The value 1 is the only duplicate.
%C A360613 All prime numbers appear in this sequence, in ascending order.
%C A360613 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?
%H A360613 Rémy Sigrist, <a href="/A360613/b360613.txt">Table of n, a(n) for n = 1..10000</a>
%H A360613 Rémy Sigrist, <a href="/A360613/a360613.txt">C program</a>
%F A360613 a(n) < a(n+2).
%e A360613 The first terms, alongside the corresponding products, are:
%e A360613   n   a(n)  Corresponding products
%e A360613   --  ----  --------------------------
%e A360613    1     1
%e A360613    2     1   1
%e A360613    3     2   2
%e A360613    4     3   3,  6
%e A360613    5     4   4, 12
%e A360613    6     5   5, 10, 20
%e A360613    7     7   7, 21, 35
%e A360613    8     8   8, 16, 32,  56
%e A360613    9     9   9, 27, 45,  72
%e A360613   10    11  11, 22, 44,  77,  99
%e A360613   11    13  13, 39, 65, 104, 143
%e A360613   12    15  15, 30, 60, 105, 135, 195
%o A360613 (C) See Links section.
%Y A360613 Cf. A000695, A062880, A066724, A360627-A360628 (bisections), A360633 (products).
%K A360613 nonn
%O A360613 1,3
%A A360613 _Rémy Sigrist_, Feb 14 2023