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

1,4

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.

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

Example

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
...

Mathematica

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

Crossrefs

Cf. A002034, A007623, A276350, A299020.

Sequence in context: A212357 A114525 A127672 * A299196 A227698 A331466

Adjacent sequences: A294165 A294166 A294167 * A294169 A294170 A294171

Keyword

nonn,base,tabf

Author

Michael De Vlieger, Feb 10 2018

Status

approved