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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.
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%I #23 Feb 23 2018 08:28:20

%S 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,

%T 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,

%U 0,1,2,3,0,0,1,2,0,2,3,6,8,9,0,9,2,7,0,16

%N 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.

%C 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.

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

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

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

%H Michael De Vlieger, <a href="/A294168/b294168.txt">Table of n, a(n) for n = 1..10384</a> (rows 1 <= n <= 250).

%H OEIS Wiki, <a href="https://oeis.org/wiki/Factorial_numeral_system">Factorial numeral system</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values">Factorial number system (Fractional values)</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%e 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.

%e 1/n expanded in factorial base appears below; this sequence includes numbers to the right of the radix point.

%e n 1/n in factorial base A276350(n) A299020(n)

%e -- ---------------------- ---------- ----------

%e 1 1.0 1 1

%e 2 0.1 1 1

%e 3 0.0 2 2 2

%e 4 0.0 1 2 3 2

%e 5 0.0 1 0 4 5 4

%e 6 0.0 1 1 1

%e 7 0.0 0 3 2 0 6 11 6

%e 8 0.0 0 3 3 3

%e 9 0.0 0 2 3 2 7 3

%e 10 0.0 0 2 2 4 2

%e 11 0.0 0 2 0 5 3 1 4 0 10 25 10

%e 12 0.0 0 2 2 2

%e 13 0.0 0 1 4 1 2 5 4 8 5 0 12 42 12

%e 14 0.0 0 1 3 3 3 10 3

%e 15 0.0 0 1 3 4 3

%e ...

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

%Y Cf. A002034, A007623, A276350, A299020.

%K nonn,base,tabf

%O 1,4

%A _Michael De Vlieger_, Feb 10 2018