A361234 -id:A361234 - cp's OEIS frontend

A361144 Lexicographically earliest sequence of positive integers such that the sums Sum_{i = 1+k2^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

Rémy Sigrist, Mar 02 2023

nonn

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).

			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

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

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

Cf. A326936, A361146, A361227.

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

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

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

Original entry on oeis.org

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

Rémy Sigrist, Mar 03 2023

nonn

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.

			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

Rémy Sigrist, PARI program

Cf. A249407, A361144, A361234.

PARI
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See Links section.
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cp's OEIS Frontend

A361144 Lexicographically earliest sequence of positive integers such that the sums Sum_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

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A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

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cp's OEIS Frontend

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.

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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.

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PARI

A361144 Lexicographically earliest sequence of positive integers such that the sums Sum_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.