A299161 - OEIS

%I #16 Mar 03 2018 09:48:49

%S 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,

%T 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,

%U 7,67,9,29,47,107,1,3,4,23,13,14,17,77,37,97,19,39,1

%N 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).

%C See A299160 for the corresponding denominators.

%C 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.

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

%C 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).

%H Rémy Sigrist, <a href="/A299161/b299161.txt">Table of n, a(n) for n = 0..10000</a>

%H Rémy Sigrist, <a href="/A299161/a299161.png">Colored logarithmic scatterplot of the first 100000 terms</a> (where the color is function of A299160(n))

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values">Factorial number system (Fractional values)</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

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

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

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

%e   n    a(n)     f(n)   fact(n)    fact(f(n))

%e   --   ----     ----   -------    ----------

%e    0      0        0         0    0.0

%e    1      1      1/2         1    0.1

%e    2      1      1/6       1 0    0.0 1

%e    3      2      2/3       1 1    0.1 1

%e    4      1      1/3       2 0    0.0 2

%e    5      5      5/6       2 1    0.1 2

%e    6      1     1/24     1 0 0    0.0 0 1

%e    7     13    13/24     1 0 1    0.1 0 1

%e    8      5     5/24     1 1 0    0.0 1 1

%e    9     17    17/24     1 1 1    0.1 1 1

%e   10      3      3/8     1 2 0    0.0 2 1

%e   11      7      7/8     1 2 1    0.1 2 1

%e   12      1     1/12     2 0 0    0.0 0 2

%e   13      7     7/12     2 0 1    0.1 0 2

%e   14      1      1/4     2 1 0    0.0 1 2

%e   15      3      3/4     2 1 1    0.1 1 2

%e   16      5     5/12     2 2 0    0.0 2 2

%e   17     11    11/12     2 2 1    0.1 2 2

%e   18      1      1/8     3 0 0    0.0 0 3

%e   19      5      5/8     3 0 1    0.1 0 3

%e   20      7     7/24     3 1 0    0.0 1 3

%t 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 *)

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

%Y Cf. A299160.

%K nonn,base,frac

%O 0,4

%A _Rémy Sigrist_, Feb 04 2018