A361144 Lexicographically earliest sequence of positive integers such that the sums Sum_{i = 1+k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct.

1, 2, 4, 5, 6, 7, 8, 10, 11, 14, 15, 17, 16, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 33, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 49, 48, 51, 52, 53, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 78, 79, 81, 80, 83, 84, 85, 86, 87, 88

Offset: 1

Author

Rémy Sigrist, Mar 02 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 (see A361227 for these values).

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 sums 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 missing value so far in the tree).

Examples

			The first terms (at the bottom of the tree) alongside the corresponding sums are:
                                176
                 ---------------------------------
                43                              133
         -----------------               -----------------
        12              31              57              76
     ---------       ---------       ---------       ---------
     3       9      13      18      25      32      35      41
   -----   -----   -----   -----   -----   -----   -----   -----
   1   2   4   5   6   7   8  10  11  14  15  17  16  19  20  21
		

Links

• Rémy Sigrist, PARI program
• Rémy Sigrist, C++ program

Crossrefs

See A360305, A361189, A361191 and A361234 for other variants.

Cf. A326936, A361146, A361227.

Programs

PARI

See Links section.
(C++) See Links section.

Formula

Empirically, a(n) ~ 4*n/3 as n tends to infinity.