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).

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

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: A092142 A348497 A376021 * A327249 A173108 A173111

Adjacent sequences: A299158 A299159 A299160 * A299162 A299163 A299164

Keyword

nonn,base,frac

Author

Rémy Sigrist, Feb 04 2018

Status

approved