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A299160
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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 denominator of f(n).
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3
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1, 2, 6, 3, 3, 6, 24, 24, 24, 24, 8, 8, 12, 12, 4, 4, 12, 12, 8, 8, 24, 24, 24, 24, 120, 120, 40, 40, 120, 120, 20, 20, 60, 60, 60, 60, 120, 120, 120, 120, 40, 40, 15, 30, 10, 5, 15, 30, 60, 60, 60, 60, 20, 20, 120, 120, 40, 40, 120, 120, 10, 5, 15, 30, 30, 15
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OFFSET
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0,2
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COMMENTS
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See A299161 for the corresponding numerators and additional comments.
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LINKS
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FORMULA
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a(n!) = (n+1)! for any n > 0.
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EXAMPLE
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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 1 0 0 0.0
1 2 1/2 1 0.1
2 6 1/6 1 0 0.0 1
3 3 2/3 1 1 0.1 1
4 3 1/3 2 0 0.0 2
5 6 5/6 2 1 0.1 2
6 24 1/24 1 0 0 0.0 0 1
7 24 13/24 1 0 1 0.1 0 1
8 24 5/24 1 1 0 0.0 1 1
9 24 17/24 1 1 1 0.1 1 1
10 8 3/8 1 2 0 0.0 2 1
11 8 7/8 1 2 1 0.1 2 1
12 12 1/12 2 0 0 0.0 0 2
13 12 7/12 2 0 1 0.1 0 2
14 4 1/4 2 1 0 0.0 1 2
15 4 3/4 2 1 1 0.1 1 2
16 12 5/12 2 2 0 0.0 2 2
17 12 11/12 2 2 1 0.1 2 2
18 8 1/8 3 0 0 0.0 0 3
19 8 5/8 3 0 1 0.1 0 3
20 24 7/24 3 1 0 0.0 1 3
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MATHEMATICA
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Block[{nn = 65, m}, m = 1; While[Factorial@ m < nn, m++]; m; {1}~Join~Denominator@ Array[NumberCompose[Prepend[#, 0], 1/Range[Length@ # + 1]!] &@ Reverse@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, nn]] (* Michael De Vlieger, Feb 10 2018 *)
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PROG
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(PARI) a(n) = my (v=0); for (r=2, oo, if (n==0, return (denominator(v))); v += (n%r)/r!; n\=r)
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CROSSREFS
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KEYWORD
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nonn,base,frac
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AUTHOR
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STATUS
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approved
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