A238757 -id:A238757 - cp's OEIS frontend

A332022 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common term in their Zeckendorf representations.

0, 2, 1, 5, 7, 3, 8, 4, 6, 13, 14, 15, 18, 9, 10, 11, 21, 23, 12, 24, 22, 16, 20, 17, 19, 34, 35, 36, 37, 38, 39, 40, 41, 47, 25, 26, 27, 28, 29, 30, 31, 32, 55, 57, 56, 60, 62, 33, 58, 59, 61, 63, 64, 65, 66, 42, 44, 43, 48, 49, 45, 50, 46, 51, 52, 53, 54, 89

Offset: 0

Rémy Sigrist, Apr 23 2020

nonn

This sequence is a self-inverse permutation of the nonnegative integers.

Apparently, {a(0), ..., a(k)} = {0, ..., k} for infinitely many integers k.

			The first terms, alongside the Zeckendorf representation in binary of n and of a(n), are:
  n   a(n)  z(n)   z(a(n))
  --  ----  -----  -------
   0     0      0        0
   1     2      1       10
   2     1     10        1
   3     5    100     1000
   4     7    101     1010
   5     3   1000      100
   6     8   1001    10000
   7     4   1010      101
   8     6  10000     1001
   9    13  10001   100000
  10    14  10010   100001

Rémy Sigrist, Table of n, a(n) for n = 0..8360
Rémy Sigrist, Scatterplot of (x, y) such that x and y have no common term in their Zeckendorf representations and 0 <= x, y <= 1218
Rémy Sigrist, PARI program for A332022
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A238757 (binary analog), A332565.

PARI
```
See Links section.
```

A003714(n) AND A003714(a(n)) = 0 for any n >= 0 (where AND denotes the bitwise AND operator).

A238758 Lexicographically earliest sequence of distinct positive integers such that a(n) AND n is a power of 2 for any n>0 (AND stands for the bitwise AND operator).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 9, 8, 7, 6, 12, 11, 17, 18, 20, 16, 13, 14, 24, 15, 26, 25, 33, 19, 22, 21, 34, 36, 35, 37, 40, 32, 23, 27, 29, 28, 30, 41, 48, 31, 38, 49, 52, 50, 65, 66, 68, 39, 42, 44, 69, 43, 67, 74, 73, 72, 71, 70, 76, 75, 80, 81

Offset: 1

Paul Tek, Mar 05 2014

nonn

This is a permutation of the positive integers.
Apparently, a self-inverse permutation.
The powers of 2 (A000079) are the fixed points.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A238757.

s = {}; Do[j=1; While[ MemberQ[s,j] || (b = BitAnd[j, n]) == 0 || BitAnd[b, b-1] > 0, j++]; AppendTo[s, j], {n, 62}]; s (* Giovanni Resta, Mar 05 2014 *)

Perl
```
See Link section.
```

A359806 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and any k > 0, floor((2^k) / n) AND floor((2^k) / a(n)) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 1, 6, 5, 4, 3, 14, 9, 8, 40, 32, 24, 60, 7, 20, 17, 16, 144, 128, 15, 72, 64, 512, 12, 256, 120, 13824, 39, 2048, 35, 62, 11, 1056, 544, 30, 288, 4096, 1008, 28, 10, 1024, 156, 5504, 1408, 112, 1424, 8192, 96, 1016, 51200, 102, 240, 32768, 27648, 248, 78

Offset: 1

Rémy Sigrist, Jan 13 2023

In other words, for any n > 0, the binary expansions of 1/n and of 1/a(n) have no common one bit; in this sense, this sequence is similar to A238757.

This sequence is a self-inverse permutation of the positive integers.

			The first terms, alongside the binary expansions of 1/n and 1/a(n) (with periodic parts in parentheses), are:
  n   a(n)  bin(1/n)        bin(1/a(n))
  --  ----  --------------  -----------
   1     2  1.(0)           0.1(0)
   2     1  0.1(0)          1.(0)
   3     6  0.(01)          0.0(01)
   4     5  0.01(0)         0.(0011)
   5     4  0.(0011)        0.01(0)
   6     3  0.0(01)         0.(01)
   7    14  0.(001)         0.0(001)
   8     9  0.001(0)        0.(000111)
   9     8  0.(000111)      0.001(0)
  10    40  0.0(0011)       0.000(0011)
  11    32  0.(0001011101)  0.00001(0)
  12    24  0.00(01)        0.000(01)

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

See A306231 for a similar sequence.

Cf. A238757.

PARI
```
See Links section.
```

A262155 Lexicographically earliest sequence of distinct positive integers such that for any n>0, n and a(n) have no common 1-bit in their binary representations, and no two successive terms have a common 1-bit in their binary representations.

Original entry on oeis.org

2, 1, 4, 3, 8, 16, 32, 5, 18, 33, 20, 34, 64, 17, 96, 6, 40, 65, 12, 35, 72, 128, 104, 7, 160, 68, 256, 66, 288, 129, 320, 9, 22, 73, 132, 10, 80, 136, 272, 67, 144, 69, 384, 19, 192, 257, 208, 11, 196, 264, 512, 74, 640, 265, 576, 130, 260, 193, 516, 131, 768

Offset: 1

Paul Tek, Sep 13 2015

This sequence combines the constraints met in A109812 and in A238757.

This sequence is a permutation of the positive integers, with inverse A262230.

			For n=5:
- the values 2, 1, 4 and 3 have already been used;
- we have the following candidates:
+---+--------+-------------+---------------+
| z | Binary | Common bits |  Common bits  |
|   | digits |   with 5    | with a(5-1)=3 |
+---+--------+-------------+---------------+
| 5 |    101 |         101 |             1 |
| 6 |    110 |         100 |            10 |
| 7 |    111 |         101 |            11 |
| 8 |   1000 |           0 |             0 |
|...|    ... |         ... |           ... |
+---+--------+-------------+---------------+
Hence, a(5)=8.

Paul Tek, Table of n, a(n) for n = 1..100000
Paul Tek, PERL program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A109812, A238757, A262230.

Perl
```
See Links section.
```

A330984 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common 1 in their base phi representations.

Original entry on oeis.org

0, 2, 1, 5, 6, 3, 4, 12, 13, 14, 16, 17, 7, 8, 9, 34, 10, 11, 30, 31, 32, 35, 36, 33, 37, 41, 42, 43, 45, 46, 18, 19, 20, 23, 15, 21, 22, 24, 88, 89, 92, 25, 26, 27, 81, 28, 29, 77, 78, 79, 82, 83, 80, 84, 90, 93, 91, 94, 106, 85, 86, 95, 96, 87, 97, 107, 108

Offset: 0

Rémy Sigrist, Apr 23 2020

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, alongside the base phi representations of n and of a(n), are:
  n   a(n)  phi(n)      phi(a(n))
  --  ----  ----------  -------------
   0     0      0            0
   1     2      1           10.01
   2     1     10.01         1
   3     5    100.01      1000.1001
   4     6    101.01      1010.0001
   5     3   1000.1001     100.01
   6     4   1010.0001     101.01
   7    12  10000.0001  100000.101001
   8    13  10001.0001  100010.001001
   9    14  10010.0101  100100.001001
  10    16  10100.0101  101000.100001

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of (x, y) such that x and y have no common 1 in their base phi representations and 0 <= x, y <= 1000
Rémy Sigrist, PARI program for A330984
Wikipedia, Golden ratio base
Index entries for sequences that are permutations of the natural numbers

Cf. A238757 (binary analog), A331212.

PARI
```
See Links section.
```

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.

Original entry on oeis.org

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
```
See Links section.
```

A353648 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) can be added without carries in balanced ternary.

Original entry on oeis.org

0, 2, 1, 5, 6, 3, 4, 17, 15, 14, 18, 16, 19, 23, 9, 8, 11, 7, 10, 12, 51, 50, 53, 13, 44, 45, 42, 41, 47, 43, 46, 54, 48, 49, 56, 52, 55, 57, 69, 68, 71, 27, 26, 29, 24, 25, 30, 28, 32, 33, 21, 20, 35, 22, 31, 36, 34, 37, 149, 153, 152, 155, 150, 151, 156, 154

Offset: 0

Rémy Sigrist, May 01 2022

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

This sequence is a self-inverse permutation of the nonnegative integers with a single fixed point: a(0) = 0.

			The first terms, in decimal and in balanced ternary, are:
  n         |  0   1   2    3    4    5    6     7     8     9    10    11    12
  a(n)      |  0   2   1    5    6    3    4    17    15    14    18    16    19
  bter(n)   |  0   1  1T   10   11  1TT  1T0   1T1   10T   100   101   11T   110
  bter(a(n))|  0  1T   1  1TT  1T0   10   11  1T0T  1TT0  1TTT  1T00  1TT1  1T01

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced ternary
Index entries for sequences that are permutations of the natural numbers

Cf. A059095, A238757 (binary analog), A353649.

ok(u, v) = { while (u && v, my (uu=[0, +1, -1][1+u%3], vv=[0, +1, -1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); 1 }
{ s=0; for (n=0, 65, for (v=0, oo, if (!bittest(s,v) && ok(n,v), print1 (v", "); s+=2^v; break))) }

A336193 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, n + a(n) can be computed without carry in base 10.

Original entry on oeis.org

1, 2, 3, 4, 10, 11, 12, 20, 30, 5, 6, 7, 13, 14, 21, 22, 31, 40, 50, 8, 15, 16, 23, 24, 32, 33, 41, 51, 60, 9, 17, 25, 26, 34, 42, 43, 52, 61, 100, 18, 27, 35, 36, 44, 53, 101, 102, 110, 120, 19, 28, 37, 45, 103, 104, 111, 112, 121, 130, 29, 38, 105, 106, 113

Offset: 1

Rémy Sigrist, Jul 11 2020

This sequence is a decimal variant of A238757.

This sequence is a self-inverse permutation of the natural numbers.

			The first terms, alongside n + a(n), are:
  n   a(n)  n+a(n)
  --  ----  ------
   1     1       2
   2     2       4
   3     3       6
   4     4       8
   5    10      15
   6    11      17
   7    12      19
   8    20      28
   9    30      39
  10     5      15
  11     6      17
  12     7      19
  13    13      26
  14    14      28
  15    21      36

Cf. A007953, A238757, A252022.

s=0; for (n=1, 64, for (v=1, oo, if (!bittest(s,v) && sumdigits(n+v)==sumdigits(n)+sumdigits(v), print1(v", "); s+=2^v; break)))

A007953(n + a(n)) = A007953(n) + A007953(a(n)).

A355166 Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.

Original entry on oeis.org

2, 1, 4, 3, 8, 17, 16, 5, 20, 21, 32, 19, 18, 33, 64, 7, 6, 13, 12, 9, 10, 41, 40, 35, 34, 37, 68, 65, 66, 97, 96, 11, 14, 25, 24, 67, 26, 73, 80, 23, 22, 85, 84, 81, 82, 129, 128, 71, 72, 69, 76, 75, 74, 137, 136, 131, 70, 133, 132, 193, 130, 257, 256, 15, 28

Offset: 1

Rémy Sigrist, Jun 22 2022

This sequence combines features of A065190 and of A238757.
This sequence is a self-inverse permutation of the nonnegative integers, without fixed points.
This sequence is well defined:
- if n is odd, then we can extend the sequence with a power of 2 > n,
- if n < 2^k is even, then we can extend the sequence with a prime number of the form 1 + t*2^k (Dirichlet's theorem on arithmetic progressions guarantees us that there is an infinity of such prime numbers).
When n is odd, a(n) is even and vice-versa.

			The first terms, alongside binary expansions and distinct prime factors, are:
  n   a(n)  bin(n)  bin(a(n))  dpf(n)  dpf(a(n))
  --  ----  ------  ---------  ------  ---------
   1     2       1         10  {}      {2}
   2     1      10          1  {2}     {}
   3     4      11        100  {3}     {2}
   4     3     100         11  {2}     {3}
   5     8     101       1000  {5}     {2}
   6    17     110      10001  {2, 3}  {17}
   7    16     111      10000  {7}     {2}
   8     5    1000        101  {2}     {5}
   9    20    1001      10100  {3}     {2, 5}
  10    21    1010      10101  {2, 5}  {3, 7}

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

Cf. A065190, A238757, A352633.

PARI
```
See Links section.
```

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    aset, mink = set(), 1
    for n in count(1):
        an = mink
        while an in aset or n&an or gcd(n, an)!=1: an += 1
        aset.add(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jun 22 2022

cp's OEIS Frontend

A332022 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common term in their Zeckendorf representations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A238758 Lexicographically earliest sequence of distinct positive integers such that a(n) AND n is a power of 2 for any n>0 (AND stands for the bitwise AND operator).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Perl

A359806 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and any k > 0, floor((2^k) / n) AND floor((2^k) / a(n)) = 0 (where AND denotes the bitwise AND operator).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A262155 Lexicographically earliest sequence of distinct positive integers such that for any n>0, n and a(n) have no common 1-bit in their binary representations, and no two successive terms have a common 1-bit in their binary representations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Perl

A330984 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common 1 in their base phi representations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A353648 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) can be added without carries in balanced ternary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A336193 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, n + a(n) can be computed without carry in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs