cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A306233 Inverse to A306231.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 12, 8, 9, 20, 17, 24, 33, 13, 22, 11, 69, 37, 51, 7, 14, 88, 109, 23, 67, 49, 232, 40, 107, 25, 91, 16, 97, 119, 26, 53, 290, 153, 41, 21, 350, 39, 253, 208, 56, 112, 669, 28, 270, 614, 30, 76, 667, 388, 95, 43, 464, 675, 671, 32, 796, 92, 45
Offset: 1

Views

Author

Rémy Sigrist, Jan 31 2019

Keywords

Examples

			A306231(39) = 42, hence a(42) = 39.

Links

Crossrefs

Cf. A306231.

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

Views

Author

Rémy Sigrist, Jan 13 2023

Keywords

Comments

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.

Examples

			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)

Links

Crossrefs

See A306231 for a similar sequence.
Cf. A238757.

Programs

  • PARI
    See Links section.

A359887 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the numerator of the unique rational q such that for any m, floor(2^m/n) AND floor(2^m/k) = floor(q*2^m) (where AND denotes the bitwise AND operator); see A359888 for the denominators.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 5, 0, 57, 1, 57, 0, 5, 0, 0, 0, 0, 1, 0, 1, 8, 8, 1, 0, 1, 0, 0, 0, 0, 85, 0, 37, 1, 1, 1, 37, 0, 85, 0, 0
Offset: 1

Views

Author

Rémy Sigrist, Jan 17 2023

Keywords

Comments

A(n, k)/A359888(n, k) can be interpreted as (1/n) AND (1/k) (assuming that inverses of powers of 2 have terminating binary expansions).

Examples

			Square array A(n, k) begins:
  n\k | 1  2   3  4    5  6     7  8       9   10      11  12
  ----+------------------------------------------------------
    1 | 1  0   0  0    0  0     0  0       0    0       0   0
    2 | 0  1   0  0    0  0     0  0       0    0       0   0
    3 | 0  0   1  1    1  0     1  0       5    1      85   1
    4 | 0  0   1  1    0  0     0  0       0    0       0   0
    5 | 0  0   1  0    1  2    57  1      37    1     837   1
    6 | 0  0   0  0    2  1     8  1       2    1       8   0
    7 | 0  0   1  0   57  8     1  1       1    1    1195   1
    8 | 0  0   0  0    1  1     1  1       0    0       0   0
    9 | 0  0   5  0   37  2     1  0       1   11  256687   5
   10 | 0  0   1  0    1  1     1  0      11    1     749   1
   11 | 0  0  85  0  837  8  1195  0  256687  749       1  85
   12 | 0  0   1  0    1  0     1  0       5    1      85   1

Links

Crossrefs

Cf. A300630, A306231, A359806, A359888 (denominators).

Programs

  • PARI
    See Links section.

Formula

A(n, k) = A(k, n).
A(n, n) = 1.
A(n, 2*n) = 0 iff n belongs to A300630.
A(A306231(n), A306231(n+1)) = 0.
A(n, A359806(n)) = 0.

A359888 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the denominator of the unique rational q such that for any m, floor(2^m/n) AND floor(2^m/k) = floor(q*2^m) (where AND denotes the bitwise AND operator); see A359887 for the numerators.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 15, 4, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 63, 1, 5, 1, 63, 1, 1, 1, 1, 1, 1, 15, 15, 1, 1, 1, 1, 1, 1, 63, 1, 455, 6, 455, 1, 63, 1, 1, 1, 1, 15, 1, 8, 63, 63, 8, 1, 15, 1, 1, 1, 1, 1023, 1, 585, 8, 7, 8, 585, 1, 1023, 1, 1
Offset: 1

Views

Author

Rémy Sigrist, Jan 17 2023

Keywords

Comments

A359887(n, k)/A(n, k) can be interpreted as (1/n) AND (1/k) (assuming that inverses of powers of 2 have terminating binary expansions).

Examples

			Square array A(n, k) begins:
  n\k | 1  2     3  4      5     6      7  8        9     10
  ----+-----------------------------------------------------
    1 | 1  1     1  1      1     1      1  1        1      1
    2 | 1  2     1  1      1     1      1  1        1      1
    3 | 1  1     3  4     15     1     63  1       63     15
    4 | 1  1     4  4      1     1      1  1        1      1
    5 | 1  1    15  1      5    15    455  8      585     15
    6 | 1  1     1  1     15     6     63  8       63     30
    7 | 1  1    63  1    455    63      7  8       63    455
    8 | 1  1     1  1      8     8      8  8        1      1
    9 | 1  1    63  1    585    63     63  1        9    117
   10 | 1  1    15  1     15    30    455  1      117     10
   11 | 1  1  1023  1  11275  1023  76461  1  3243933  11275
   12 | 1  1    12  1     15     1     63  1       63     15

Links

Crossrefs

Cf. A300630, A306231, A359806, A359887 (numerators).

Programs

  • PARI
    See Links section.

Formula

A(n, k) = A(k, n).
A(n, n) = n.
Showing 1-4 of 4 results.

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