login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A299161 In factorial base, any rational number has a terminating expansion; hence we can devise a self-inverse permutation of the rational numbers, say f, such that for any rational number q, the representations of q and of f(q) in factorial base are mirrored around the radix point and q and f(q) have the same sign; a(n) = the numerator of f(n). 3
0, 1, 1, 2, 1, 5, 1, 13, 5, 17, 3, 7, 1, 7, 1, 3, 5, 11, 1, 5, 7, 19, 11, 23, 1, 61, 7, 27, 41, 101, 1, 11, 13, 43, 23, 53, 11, 71, 31, 91, 17, 37, 2, 19, 3, 4, 7, 29, 1, 31, 11, 41, 7, 17, 7, 67, 9, 29, 47, 107, 1, 3, 4, 23, 13, 14, 17, 77, 37, 97, 19, 39, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

See A299160 for the corresponding denominators.

The function f restricted to the nonnegative integers establishes a bijection from the nonnegative integers to the rational numbers q such that 0 <= q < 1, hence n -> a(n) / A299161(n) runs uniquely through all rational numbers q such that 0 <= q < 1.

The rational numbers q = n + f(n) for some integer n are the fixed points of f.

If two rational numbers, say p and q, have the same sign and can be added without carry in factorial base, then f(p + q) = f(p) + f(q).

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A299160(n))

Wikipedia, Factorial number system (Fractional values)

Index entries for sequences related to factorial base representation

FORMULA

a(n) < A299160(n) for any n >= 0.

a(n!) = 1 for any n >= 0.

EXAMPLE

The first terms, alongside f(n) and the factorial base representations of n and of f(n), are:

  n    a(n)     f(n)   fact(n)    fact(f(n))

  --   ----     ----   -------    ----------

   0      0        0         0    0.0

   1      1      1/2         1    0.1

   2      1      1/6       1 0    0.0 1

   3      2      2/3       1 1    0.1 1

   4      1      1/3       2 0    0.0 2

   5      5      5/6       2 1    0.1 2

   6      1     1/24     1 0 0    0.0 0 1

   7     13    13/24     1 0 1    0.1 0 1

   8      5     5/24     1 1 0    0.0 1 1

   9     17    17/24     1 1 1    0.1 1 1

  10      3      3/8     1 2 0    0.0 2 1

  11      7      7/8     1 2 1    0.1 2 1

  12      1     1/12     2 0 0    0.0 0 2

  13      7     7/12     2 0 1    0.1 0 2

  14      1      1/4     2 1 0    0.0 1 2

  15      3      3/4     2 1 1    0.1 1 2

  16      5     5/12     2 2 0    0.0 2 2

  17     11    11/12     2 2 1    0.1 2 2

  18      1      1/8     3 0 0    0.0 0 3

  19      5      5/8     3 0 1    0.1 0 3

  20      7     7/24     3 1 0    0.0 1 3

MATHEMATICA

Block[{nn = 72, m}, m = 1; While[Factorial@ m < nn, m++]; m; {0}~Join~Numerator@ Array[NumberCompose[Prepend[#, 0], 1/Range[Length@ # + 1]!] &@Reverse@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, nn]] (* Michael De Vlieger, Feb 10 2018 *)

PROG

(PARI) a(n) = my (v=0); for (r=2, oo, if (n==0, return (numerator(v))); v += (n%r)/r!; n\=r)

CROSSREFS

Cf. A299160.

Sequence in context: A163963 A119763 A092142 * A173108 A173111 A257459

Adjacent sequences:  A299158 A299159 A299160 * A299162 A299163 A299164

KEYWORD

nonn,base,frac

AUTHOR

Rémy Sigrist, Feb 04 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 13 13:15 EST 2018. Contains 317149 sequences. (Running on oeis4.)