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 1-3 of 3 results.

A337223 a(n) is the least number that can be obtained by replacing some cube XXX in the binary expansion of n by X.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 2, 9, 10, 11, 12, 13, 2, 3, 4, 5, 18, 19, 20, 21, 22, 5, 6, 25, 26, 27, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 9, 10, 41, 2, 43, 44, 45, 10, 11, 12, 13, 50, 51, 52, 53, 54, 13, 8, 9, 10, 11, 12, 13, 14, 3, 4, 17, 18, 19, 20, 21, 22, 17
Offset: 0

Views

Author

Rémy Sigrist, Aug 19 2020

Keywords

Comments

Leading zeros in binary expansions are ignored.
Fixed points correspond to A286262.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     3      11         11
   4     4     100        100
   5     5     101        101
   6     6     110        110
   7     1     111          1
   8     2    1000         10
   9     9    1001       1001
  10    10    1010       1010
  11    11    1011       1011
  12    12    1100       1100
  13    13    1101       1101
  14     2    1110         10
  15     3    1111         11
  16     4   10000        100

Links

Crossrefs

Cf. A286262, A297405, A337222, A337224.

Programs

  • PARI
    See Links section.

Formula

a(A297405(n)) = n for any n > 0.

A337224 a(n) is the least number that can be obtained by replacing some repetitive part X^k in the binary expansion of n by X.

Original entry on oeis.org

0, 1, 2, 1, 2, 5, 2, 1, 2, 5, 2, 5, 4, 5, 2, 1, 2, 5, 10, 9, 4, 5, 10, 5, 6, 9, 6, 11, 4, 5, 2, 1, 2, 5, 10, 11, 4, 9, 18, 9, 8, 9, 2, 11, 20, 5, 10, 5, 6, 13, 18, 19, 12, 13, 6, 13, 8, 9, 10, 11, 4, 5, 2, 1, 2, 5, 10, 11, 20, 17, 22, 17, 8, 9, 18, 19, 36, 37
Offset: 0

Views

Author

Rémy Sigrist, Aug 19 2020

Keywords

Comments

Leading zeros in binary expansions are ignored.
There are four fixed points: 0, 1, 2 and 5; their binary expansion is a squarefree string.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     1      11          1
   4     2     100         10
   5     5     101        101
   6     2     110         10
   7     1     111          1
   8     2    1000         10
   9     5    1001        101
  10     2    1010         10
  11     5    1011        101
  12     4    1100        100
  13     5    1101        101
  14     2    1110         10
  15     1    1111          1
  16     2   10000         10

Links

Crossrefs

Cf. A337222, A337223.

Programs

  • PARI
    See Links section.

Formula

a(n) = 1 iff n = 2^k-1 for some k > 0.

A337225 a(n) is the number of distinct integers k that can be obtained by starting from the binary expansion of n and repeatedly replacing some square XX by X.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 2, 4, 2, 3, 4, 4, 3, 3, 4, 4, 2, 3, 3, 6, 4, 4, 4, 6, 3, 4, 5, 5, 4, 4, 6, 6, 4, 5, 6, 6, 4, 3, 4, 6, 3, 4, 4, 8, 6, 6, 8, 8, 4, 6, 6, 9, 6, 6, 6, 8, 4, 5, 6, 6, 5, 5, 8, 8, 6, 7, 9, 9, 6, 5, 8, 10, 6, 7, 8, 8, 6, 5, 8, 6, 3, 5
Offset: 0

Views

Author

Rémy Sigrist, Aug 19 2020

Keywords

Comments

Leading zeros in binary expansions are ignored.
The least possible k is:
- 0 for n = 0,
- 1 for n = 2^m-1 for some m > 0,
- 2 for n = 2*m for some m > 0,
- 5 otherwise.

Examples

			The first terms, alongside the binary expansions of n and of the corresponding k's, are:
  n   a(n)  bin(n)  {bin(k)}
  --  ----  ------  -------------------
   0     1       0  {0}
   1     1       1  {1}
   2     1      10  {10}
   3     2      11  {1, 11}
   4     2     100  {10, 100}
   5     1     101  {101}
   6     2     110  {10, 110}
   7     3     111  {1, 11, 111}
   8     3    1000  {10, 100, 1000}
   9     2    1001  {101, 1001}
  10     2    1010  {10, 1010}
  11     2    1011  {101, 1011}
  12     4    1100  {10, 100, 110, 1100}
  13     2    1101  {101, 1101}
  14     3    1110  {10, 110, 1110}
  15     4    1111  {1, 11, 111, 1111}
  16     4   10000  {10, 100, 1000, 10000}

Links

Crossrefs

Cf. A337222.

Programs

  • PARI
    See Links section.

Formula

a(2^k-1) = k for any k > 0.
Showing 1-3 of 3 results.

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