A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct.

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Author

Rémy Sigrist, Mar 03 2023

Keywords

• nonn

Comments

In other words, a(1), a(2), a(1)*a(2), a(3), a(4), a(3)*a(4), a(1)*a(2)*a(3)*a(4), a(5), a(6), a(5)*a(6), etc. are all distinct.

In particular, all terms are distinct (but not necessarily in increasing order).

We can arrange the terms of the sequence as the leaves of a perfect infinite binary tree, the products with e > 0 corresponding to parent nodes; each node will contain a different value and all values will appear in the tree (if n = 2^m+1 for some m > 0, then a(n) will equal the least value > 1 missing so far in the tree).

This sequence is a variant of A361144 where we use products instead of sums.

The data section matches that of A249407, however a(115) = 121 whereas A249407(115) = 120.

Examples

			The first terms (at the bottom of the tree) alongside the corresponding products are:
                          1067062284288000
                  ---------------------------------
               604800                        1764322560
          -----------------               -----------------
         120            5040            24024           73440
      ---------       ---------       ---------       ---------
      6      20      56      90      132     182     240     306
    -----   -----   -----   -----   -----   -----   -----   -----
    2   3   4   5   7   8   9  10  11  12  13  14  15  16  17  18
		

Links

• Rémy Sigrist, PARI program

Crossrefs

Cf. A249407, A361144, A361234.

Programs

PARI

See Links section.