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 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) 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

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