A323626 For any nonnegative real number x, let f(x) be the real number obtained by replacing in the binary expansions of the integer and fractional parts of x each finite run of k consecutive equal bits b with a run of k-(-1)^k consecutive bits b; a(n) is the numerator of f(1/n).

3, 3, 1, 3, 1, 2, 3, 3, 1, 1, 13, 1, 7, 3, 1, 3, 1, 2, 77, 1, 1, 26, 203, 1, 817, 14, 109, 3, 1037, 2, 3, 3, 1, 1, 1297, 1, 20275, 77, 155, 1, 17, 1, 13, 13, 67, 203, 6716227, 1, 421735, 817, 17, 7, 2306997, 109, 55739, 3, 49, 1037, 818712813, 1, 138203853, 3

Offset: 1

Author

Rémy Sigrist, Jan 20 2019

Keywords

• nonn
• frac
• base

Comments

When computing f(x), we consider the unique binary representation of x where the fractional part of x does not eventually end with repeating ones.

The function f establishes a self-inverse bijection:

- over the nonnegative real numbers,

- over the nonnegative real numbers in the half-open interval [0,1),

- over the nonnegative rational numbers,

- over the nonnegative rational numbers in the half-open interval [0,1),

- over the nonnegative integers (for any n >= 0, f(n) = A162853(n)).

The function f has only one fixed point: f(0) = 0.

Examples

			The first terms of the sequence, alongside f(1/n) and the binary representations of 1/n and of f(1/n) with periodic part in parentheses, are:
  n   a(n)  f(1/n)   bin(1/n)                bin(f(1/n))
  --  ----  -------  ----------------------  ------------------------
   1     3        3  1.(0)                   11.(0)
   2     3      3/4  0.1(0)                   0.11(0)
   3     1      1/5  0.(01)                   0.(0011)
   4     3     3/16  0.01(0)                  0.0011(0)
   5     1      1/3  0.(0011)                 0.(01)
   6     2      2/5  0.0(01)                  0.(0110)
   7     3      3/7  0.(001)                  0.(011)
   8     3      3/8  0.001(0)                 0.011(0)
   9     1     1/17  0.(000111)               0.(00001111)
  10     1     1/24  0.0(0011)                0.000(01)
  11    13   13/257  0.(0001011101)           0.(0000110011110011)
  12     1     1/20  0.00(01)                 0.00(0011)
  13     7    7/129  0.(000100111011)         0.(00001101111001)
  14     3     3/56  0.0(001)                 0.000(011)
  15     1     1/21  0.(0001)                 0.(000011)
  16     3     3/64  0.0001(0)                0.000011(0)
  17     1      1/9  0.(00001111)             0.(000111)
  18     2     2/17  0.0(000111)              0.(00011110)
  19    77  77/1025  0.(000011010111100101)   0.(00010011001110110011)
  20     1     1/12  0.00(0011)               0.00(01)
		

Links

• Rémy Sigrist, Representation of f in the half-open interval [0,1)
• Rémy Sigrist, PARI program for A323626

Crossrefs

See A323627 for the corresponding denominators.

Cf. A162853.

Programs

PARI

See Links section.

Formula

a(2^k) = 3 for any k >= 0.

a(2^k-1) = 2-(-1)^k for any k > 0.