A360305 -id:A360305 - 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.

A361234 Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

Original entry on oeis.org

-1, 2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, 10, -10, 11, -11, 12, -12, 13, -13, 14, -14, 15, -15, 16, 17, -17, -18, 19, -19, 20, -20, 21, -21, 22, -22, 23, -23, 24, -24, 25, 26, -26, 27, -27, 28, -28, 29, -29, 30, -30, 31, -31, 32, -32, 33, -33, 34

Offset: 1

Rémy Sigrist, Mar 05 2023

sign

This sequence is a variant of A360305 where we allow negative values.
In order for the sequence to be infinite, the value 1 is forbidden.
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 except 0 and 1 will appear in the tree.

			The first terms (at the bottom of the tree) alongside the corresponding products are:
                           -73156608000
                 ---------------------------------
               7200                          -10160640
         -----------------               -----------------
        18              400            1764            -5760
     ---------       ---------       ---------       ---------
    -2      -9      -16     -25     -36     -49     -64     90
   -----   -----   -----   -----   -----   -----   -----   -----
  -1   2   3  -3   4  -4   5  -5   6  -6   7  -7   8  -8   9  10

Rémy Sigrist, PARI program

Cf. A360305, A361144.

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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A361234 Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

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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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A361234 Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

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.

A361234 Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.