A299020 a(n) is the maximum digit in the factorial base expansion of 1/n.

1, 1, 2, 2, 4, 1, 6, 3, 3, 2, 10, 2, 12, 3, 3, 3, 16, 4, 18, 1, 4, 6, 22, 1, 7, 9, 5, 5, 28, 4, 30, 4, 7, 9, 4, 3, 36, 13, 8, 3, 40, 5, 42, 8, 4, 15, 46, 3, 11, 6, 12, 9, 52, 6, 8, 6, 15, 15, 58, 2, 60, 22, 5, 6, 7, 9, 66, 12, 17, 4, 70, 4, 72, 31, 5, 14, 7

Offset: 1

Author

Rémy Sigrist, Jan 31 2018

Keywords

• nonn
• base

Comments

See the Wikipedia link for the construction method of 1/n in factorial base.

Examples

			The first terms, alongside 1/n in factorial base, are:
  n     a(n)      1/n in factorial base
  --    ----      ---------------------
   1       1      1
   2       1      0.1
   3       2      0.0 2
   4       2      0.0 1 2
   5       4      0.0 1 0 4
   6       1      0.0 1
   7       6      0.0 0 3 2 0 6
   8       3      0.0 0 3
   9       3      0.0 0 2 3 2
  10       2      0.0 0 2 2
  11      10      0.0 0 2 0 5 3 1 4 0 10
  12       2      0.0 0 2
  13      12      0.0 0 1 4 1 2 5 4 8 5 0 12
  14       3      0.0 0 1 3 3 3
  15       3      0.0 0 1 3
  16       3      0.0 0 1 2 3
  17      16      0.0 0 1 2 0 2 3 6 8 9 0 9 2 7 0 16
  18       4      0.0 0 1 1 4
  19      18      0.0 0 1 1 1 6 2 0 9 5 2 6 11 11 13 8 0 18
  20       1      0.0 0 1 1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of the first 25000 terms (where the color is function of A052126(n))
• Wikipedia, Factorial number system (Fractional values)
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A052126, A246359, A276350.

Programs

Mathematica

a[n_] := Module[{m = 0, r = 1, f = 1/n}, While[f > 0, m = Max[m, Floor[f]]; r++; f = FractionalPart[f]*r]; m]; Array[a, 77] (* Jean-François Alcover, Feb 05 2018, translated from PARI *)

PARI

a(n) = my (m=0, r=1, f=1/n); while (f>0, m = max(m, floor(f)); r++; f = frac(f)*r); return (m)

Formula

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

a(n! / k) = k for any n > 1 and k = 1..n-1.

a(p) = p - 1 for any prime p.