A299161 -id:A299161

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A299160

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).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

See A299161 for the corresponding numerators and additional comments.

LINKS

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

Wikipedia, Factorial number system (Fractional values)

Index entries for sequences related to factorial base representation

FORMULA

a(n!) = (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 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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A299161.

KEYWORD

nonn,base,frac

AUTHOR

Rémy Sigrist, Feb 04 2018

STATUS

approved

A299199

+10
1

1, 4, 14, 98, 2, 4386, 18, 324, 60, 36457092, 12, 5769254382, 2598, 78, 414, 335391687123174, 510, 115428139222691670, 30, 1926, 20204166, 24752828962220504429646, 6, 1032336, 3124309416, 149376, 3816, 8542182056001396008878674488976, 96

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

See A299161 for additional comments about f.

This sequence corresponds to the indices of ones in A299161.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 2..456

Wikipedia, Factorial number system (Fractional values)

Index entries for sequences related to factorial base representation

Rémy Sigrist, Colored logarithmic scatterplot of the first 5000 terms (where the color is function of A299160(n))

FORMULA

A034968(a(n)) = A276350(n) for any n > 1.

A299160(a(n)) = n for any n > 1.

A299161(a(n)) = 1 for any n > 1.

EXAMPLE

The first terms, alongside the factorial base representations of a(n) and of 1/n, are:

n a(n) fact(a(n)) fact(1/n)

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

2 1 1 0.1

3 4 2 0 0.0 2

4 14 2 1 0 0.0 1 2

5 98 4 0 1 0 0.0 1 0 4

6 2 1 0 0.0 1

7 4386 6 0 2 3 0 0 0.0 0 3 2 0 6

8 18 3 0 0 0.0 0 3

9 324 2 3 2 0 0 0.0 0 2 3 2

10 60 2 2 0 0 0.0 0 2 2

11 36457092 10 0 4 1 3 5 0 2 0 0 0.0 0 2 0 5 3 1 4 0 10

12 12 2 0 0 0.0 0 2

13 5769254382 12 0 5 8 4 5 2 1 4 1 0 0 0.0 0 1 4 1 2 5 4 8 5 0 12

14 2598 3 3 3 1 0 0 0.0 0 1 3 3 3

15 78 3 1 0 0 0.0 0 1 3

16 414 3 2 1 0 0 0.0 0 1 2 3

PROG

(PARI) a(n) = my (v=0, q=1/n); for (r=2, oo, q *= r; v += floor(q) * (r-1)!; q = frac(q); if (q==0, return (v)))