A352724 -id:A352724 - cp's OEIS frontend

A352726 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and a(n) have no common runs of consecutive 1's.

0, 2, 1, 4, 3, 6, 5, 8, 7, 12, 13, 14, 9, 10, 11, 16, 15, 24, 25, 26, 27, 28, 29, 30, 17, 18, 19, 20, 21, 22, 23, 32, 31, 48, 49, 50, 51, 54, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 33, 34, 35, 36, 38, 39, 37, 40, 41, 42, 43, 44, 45, 46, 47, 64, 63, 96, 97, 98

Offset: 0

Rémy Sigrist, Mar 30 2022

This sequence is a self-inverse permutation of the nonnegative integers.
This sequence has similarities with A238757; here we consider runs of consecutive 1's, there individual 1's in binary expansions.
The binary expansion of n and a(n) may share some 1's, but cannot have a common run of consecutive 1's (as given by A352724).

			The first terms, alongside the corresponding partitions into runs of 1's, are:
  n   a(n)  runs in n  runs in a(n)
  --  ----  ---------  ------------
   0     0  []         []
   1     2  [1]        [2]
   2     1  [2]        [1]
   3     4  [3]        [4]
   4     3  [4]        [3]
   5     6  [1, 4]     [6]
   6     5  [6]        [1, 4]
   7     8  [7]        [8]
   8     7  [8]        [7]
   9    12  [1, 8]     [12]
  10    13  [2, 8]     [1, 12]
  11    14  [3, 8]     [14]
  12     9  [12]       [1, 8]
  13    10  [1, 12]    [2, 8]
  14    11  [14]       [3, 8]
  15    16  [15]       [16]
  16    15  [16]       [15]

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 32769 terms
Rémy Sigrist, Scatterplot of (x, y) such that x, y < 2^10 and the binary expansions of x and y have no common runs of consecutive 1's
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A238757, A332022, A352724.

PARI
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See Links section.
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A352727 Square array A(n, k), n, k >= 0, read by antidiagonals: the binary expansion of A(n, k) contains the runs of consecutive 1's that appear both in the binary expansions of n and k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0

Offset: 0

Rémy Sigrist, Mar 30 2022

We only consider maximal runs of one or more consecutive 1's (as counted by A069010) that completely match in binary expansions of n and k, not simply single common 1's.

			Table A(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+------------------------------------------------------
    0|  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
    1|  0  1  0  0  0  1  0  0  0  1   0   0   0   1   0   0
    2|  0  0  2  0  0  0  0  0  0  0   2   0   0   0   0   0
    3|  0  0  0  3  0  0  0  0  0  0   0   3   0   0   0   0
    4|  0  0  0  0  4  4  0  0  0  0   0   0   0   0   0   0
    5|  0  1  0  0  4  5  0  0  0  1   0   0   0   1   0   0
    6|  0  0  0  0  0  0  6  0  0  0   0   0   0   0   0   0
    7|  0  0  0  0  0  0  0  7  0  0   0   0   0   0   0   0
    8|  0  0  0  0  0  0  0  0  8  8   8   8   0   0   0   0
    9|  0  1  0  0  0  1  0  0  8  9   8   8   0   1   0   0
   10|  0  0  2  0  0  0  0  0  8  8  10   8   0   0   0   0
   11|  0  0  0  3  0  0  0  0  8  8   8  11   0   0   0   0
   12|  0  0  0  0  0  0  0  0  0  0   0   0  12  12   0   0
   13|  0  1  0  0  0  1  0  0  0  1   0   0  12  13   0   0
   14|  0  0  0  0  0  0  0  0  0  0   0   0   0   0  14   0
   15|  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k); black pixels denote 0's)
Index entries for sequences related to binary expansion of n

Cf. A004198, A069010, A352724, A352729.

A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }
A(n,k) = vecsum(setintersect(A352724(n), A352724(k)))

A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, n) = n.
A(n, 2*n) = 0.
A(n, k) <= A004198(n, k) (bitwise AND operator).
A(n, n+1) = A352729(n).

A352729 The binary expansion of a(n) contains the runs of consecutive 1's that appear both in the binary expansions of n and n+1.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 8, 8, 8, 0, 12, 0, 0, 0, 16, 16, 16, 16, 20, 16, 16, 0, 24, 24, 24, 0, 28, 0, 0, 0, 32, 32, 32, 32, 36, 32, 32, 32, 40, 40, 40, 32, 44, 32, 32, 0, 48, 48, 48, 48, 52, 48, 48, 0, 56, 56, 56, 0, 60, 0, 0, 0, 64, 64, 64, 64, 68, 64, 64, 64

Offset: 0

Rémy Sigrist, Mar 30 2022

We only consider runs of consecutive 1's that completely match in binary expansions of n and n+1, not simply single common 1's.

			For n = 42:
- the binary expansion of 42 is "101010",
- the binary expansion of 43 is "101011",
- the first two runs of 1's are the same, the others differ,
- so the binary expansion of a(42) is "101000",
- and a(42) = 40.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A129760, A352727.

A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }
a(n) = vecsum(setintersect(A352724(n), A352724(n+1)))

a(n) = A352727(n, n+1).

A352725 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of a(n) and a(n+1) have no common runs of consecutive 1's.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 14, 10, 13, 11, 15, 16, 24, 17, 26, 19, 25, 18, 27, 20, 28, 21, 30, 22, 29, 23, 31, 32, 48, 33, 50, 35, 49, 34, 51, 36, 54, 37, 55, 38, 52, 39, 53, 40, 56, 41, 58, 43, 57, 42, 59, 44, 60, 45, 62, 46, 61, 47, 63, 64, 96, 65, 98

Offset: 0

Rémy Sigrist, Mar 30 2022

This sequence is a variant of A109812; here we consider runs of consecutive 1's, there individual 1's in binary expansions.

The binary expansions of two consecutive terms may share some 1's, but cannot have a common run of consecutive 1's (as given by A352724).

			The first terms, alongside the corresponding partitions into runs of 1's, are:
  n   a(n)  runs in a(n)
  --  ----  ------------
   0     0  []
   1     1  [1]
   2     2  [2]
   3     3  [3]
   4     4  [4]
   5     6  [6]
   6     5  [1, 4]
   7     7  [7]
   8     8  [8]
   9    12  [12]
  10     9  [1, 8]
  11    14  [14]
  12    10  [2, 8]
  13    13  [1, 12]
  14    11  [3, 8]
  15    15  [15]
  16    16  [16]

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 32769 terms
Rémy Sigrist, PARI program

Cf. A109812, A332565, A352724.

PARI
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See Links section.
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cp's OEIS Frontend

A352726 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and a(n) have no common runs of consecutive 1's.

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A352727 Square array A(n, k), n, k >= 0, read by antidiagonals: the binary expansion of A(n, k) contains the runs of consecutive 1's that appear both in the binary expansions of n and k.

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A352729 The binary expansion of a(n) contains the runs of consecutive 1's that appear both in the binary expansions of n and n+1.

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PARI

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A352725 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of a(n) and a(n+1) have no common runs of consecutive 1's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI