A361144 -id:A361144 - cp's OEIS frontend

A361146 a(n) is the sibling of n in the infinite binary tree underlying A361144.

2, 1, 9, 5, 4, 7, 6, 10, 3, 8, 14, 31, 18, 11, 17, 19, 15, 13, 16, 21, 20, 23, 22, 26, 32, 24, 28, 27, 30, 29, 12, 25, 34, 33, 41, 37, 36, 39, 38, 42, 35, 40, 133, 46, 50, 44, 49, 51, 47, 45, 48, 53, 52, 56, 59, 54, 76, 60, 55, 58, 62, 61, 64, 63, 66, 65, 73

Offset: 1

Rémy Sigrist, Mar 02 2023

nonn

This sequence is a self-inverse permutation of the positive integers with no fixed point.

			The infinite binary tree underlying A361144 starts as follows:
                                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
.
So a(1) = 2 and a(2) = 1, a(4) = 5 and a(5) = 4, etc.,
   a(3) = 9 and a(9) = 3, a(13) = 18 and a(18) = 13, etc.,
   a(12) = 31 and a(31) = 12, a(57) = 76 and a(76) = 57,
   a(43) = 133 and a(133) = 43.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A328654, A361144.

PARI
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See Links section.
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a(A361144(2*n)) = A361144(2*n-1).

a(A361144(2*n-1)) = A361144(2*n).

A361227 Irregular triangle T(n, k), n > 0, k = 0..A007814(n), read by rows: T(n, k) = Sum_{i = n-2^k+1..n} A361144(i).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 12, 6, 7, 13, 8, 10, 18, 31, 43, 11, 14, 25, 15, 17, 32, 57, 16, 19, 35, 20, 21, 41, 76, 133, 176, 22, 23, 45, 24, 26, 50, 95, 27, 28, 55, 29, 30, 59, 114, 209, 33, 34, 67, 36, 37, 73, 140, 38, 39, 77, 40, 42, 82, 159, 299, 508, 684, 44, 46, 90

Offset: 1

Rémy Sigrist, Mar 05 2023

This sequence gives the sums underlying A361144: the n-th row corresponds to the sums where A361144(n) is the last term.

Each integer appears once in this sequence.

			Triangle T(n, k) begins:
  n   n-th row
  --  ----------------
   1   1
   2   2,  3
   3   4
   4   5,  9, 12
   5   6
   6   7, 13
   7   8
   8  10, 18, 31, 43
   9  11
  10  14, 25
  11  15
  12  17, 32, 57

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, C++ program
Index entries for sequences that are permutations of the natural numbers

Cf. A007814, A361144.

T(n, 0) = A361144(n).

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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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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A361189 Infinite sequence of nonzero integers build the greedy way such that the sums Sum_{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, -1, -4, -3, -6, 4, -11, 5, 6, 7, 8, -8, -12, 9, 21, -10, -13, 12, 25, 13, 16, -14, 31, -15, -17, 19, 33, -19, -21, 22, 41, -22, -24, 24, 49, -25, -26, -27, -28, 28, 34, -29, 61, -30, -31, -33, -34, 35, 39, -35, 75, -36, -37, 38, 77, -38, -39, -41, -42

Offset: 1

Rémy Sigrist, Mar 03 2023

sign

This sequence is a variant of A361144 where we allows negative values.
In order for the sequence to be infinite, zero sums are forbidden.
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 nonzero values will appear in the tree.

			The first terms (at the bottom of the tree) alongside the corresponding sums are:
                                18
                 ---------------------------------
                -18                             36
         -----------------               -----------------
        -2              -16             26              10
     ---------       ---------       ---------       ---------
     3      -5      -9      -7      11      15      -20     30
   -----   -----   -----   -----   -----   -----   -----   -----
   1   2  -1  -4  -3  -6   4 -11   5   6   7   8  -8 -12   9  21

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program

Cf. A361144.

A361191 Lexicographically earliest sequence of positive integers such that the sums SumXOR_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct (where SumXOR is the analog of summation under the binary XOR operation).

Original entry on oeis.org

1, 2, 4, 8, 5, 11, 6, 16, 7, 10, 9, 21, 18, 32, 19, 64, 20, 33, 25, 49, 26, 34, 27, 65, 30, 35, 31, 66, 36, 71, 37, 105, 38, 67, 39, 108, 41, 68, 42, 128, 43, 69, 44, 116, 45, 70, 51, 176, 52, 72, 57, 129, 58, 73, 59, 118, 60, 78, 63, 130, 74, 132, 80, 256, 81

Offset: 1

Rémy Sigrist, Mar 03 2023

In other words, a(1), a(2), a(1) XOR a(2), a(3), a(4), a(3) XOR a(4), a(1) XOR a(2) XOR a(3) XOR a(4), a(5), a(6), a(5) XOR 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 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).
This sequence is a variant of A361144 based on the bitwise XOR operator.

			The first terms (at the bottom of the tree) alongside the corresponding sums are:
                                 103
                  ---------------------------------
                 23                              112
          -----------------               -----------------
         15              24              17              97
      ---------       ---------       ---------       ---------
      3      12      14      22      13      28      50      83
    -----   -----   -----   -----   -----   -----   -----   -----
    1   2   4   8   5  11   6  16   7  10   9  21  18  32  19  64

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, C++ program

Cf. A361144.

cp's OEIS Frontend

A361146 a(n) is the sibling of n in the infinite binary tree underlying A361144.

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A361227 Irregular triangle T(n, k), n > 0, k = 0..A007814(n), read by rows: T(n, k) = Sum_{i = n-2^k+1..n} A361144(i).

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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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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.

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PARI

A361189 Infinite sequence of nonzero integers build the greedy way such that the sums Sum_{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.

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A361191 Lexicographically earliest sequence of positive integers such that the sums SumXOR_{i = 1+k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct (where SumXOR is the analog of summation under the binary XOR operation).

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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.

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.

A361189 Infinite sequence of nonzero integers build the greedy way such that the sums Sum_{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.

A361191 Lexicographically earliest sequence of positive integers such that the sums SumXOR_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct (where SumXOR is the analog of summation under the binary XOR operation).