A294168 Irregular triangle read by rows in which row n contains significant digits after the radix point for unit fractions 1/n expanded in factorial base.

0, 1, 0, 2, 0, 1, 2, 0, 1, 0, 4, 0, 1, 0, 0, 3, 2, 0, 6, 0, 0, 3, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 0, 2, 0, 5, 3, 1, 4, 0, 10, 0, 0, 2, 0, 0, 1, 4, 1, 2, 5, 4, 8, 5, 0, 12, 0, 0, 1, 3, 3, 3, 0, 0, 1, 3, 0, 0, 1, 2, 3, 0, 0, 1, 2, 0, 2, 3, 6, 8, 9, 0, 9, 2, 7, 0, 16

Offset: 1

Author

Michael De Vlieger, Feb 10 2018

Keywords

• nonn
• base
• tabf

Comments

See the Wikipedia link for the construction method of 1/n in factorial base. This version eliminates the 1/0! and 1/1! places, which are always 0.

By convention, row n = 1 contains {0}.

Length of row n = A002034(n) - 1.

Length of row p = p - 1 for p prime.

Examples

			Row n = 7 contains {0, 0, 3, 2, 0, 6} since these are the digits that appear after the radix point in the factorial base expansion of 1/7. The length of row 7 = A002034(7) - 1 = 6, the largest number in row 7 is A299020(7) = 6, and the sum of row 7 = A276350(7) = 11.
1/n expanded in factorial base appears below; this sequence includes numbers to the right of the radix point.
   n    1/n in factorial base     A276350(n)  A299020(n)
  --    ----------------------    ----------  ----------
   1    1.0                            1          1
   2    0.1                            1          1
   3    0.0 2                          2          2
   4    0.0 1 2                        3          2
   5    0.0 1 0 4                      5          4
   6    0.0 1                          1          1
   7    0.0 0 3 2 0 6                 11          6
   8    0.0 0 3                        3          3
   9    0.0 0 2 3 2                    7          3
  10    0.0 0 2 2                      4          2
  11    0.0 0 2 0 5 3 1 4 0 10        25         10
  12    0.0 0 2                        2          2
  13    0.0 0 1 4 1 2 5 4 8 5 0 12    42         12
  14    0.0 0 1 3 3 3                 10          3
  15    0.0 0 1 3                      4          3
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10384 (rows 1 <= n <= 250).
• OEIS Wiki, Factorial numeral system
• Wikipedia, Factorial number system (Fractional values)
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A002034, A007623, A276350, A299020.

Programs

Mathematica

Array[With[{s = NumberDecompose[1/#, 1/Range[#]!]}, Rest@ Drop[s, -LengthWhile[Reverse@ s, # == 0 &]]] - Boole[# == 1] &, 17] /. {} -> {0} // Flatten