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A373085 Numbers k such that the factorial base representation of 1/k without the leading zeros is palindromic.

%I A373085 #8 May 26 2024 16:11:04
%S A373085 1,2,3,6,8,9,10,12,20,24,30,40,60,120,126,144,160,180,189,210,240,315,
%T A373085 360,384,630,720,840,896,1008,1056,1120,1260,1680,2240,2520,4480,5040,
%U A373085 5184,5760,6048,6300,6720,6912,8064,9072,9450,10080,12096,13440,14400,18144
%N A373085 Numbers k such that the factorial base representation of 1/k without the leading zeros is palindromic.
%C A373085 All the factorials (A000142) are terms, since the factorial base representation of 1/k! is k-1 0's followed by 1.
%C A373085 If k > 4 is composite then (k-1)!/k is a term.
%H A373085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values">Factorial number system (Fractional values)</a>.
%H A373085 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%e A373085 The first 10 terms are:
%e A373085    n  a(n)   1/a(n) in factorial base
%e A373085   --  ----   ------------------------
%e A373085    1    1    1.
%e A373085    2    2    0.1
%e A373085    3    3    0.02
%e A373085    4    6    0.01
%e A373085    5    8    0.003
%e A373085    6    9    0.00232
%e A373085    7   10    0.0022
%e A373085    8   12    0.002
%e A373085    9   20    0.0011
%e A373085   10   24    0.001
%t A373085 q[n_] := Module[{d = NumberDecompose[1/n, 1/Range[n]!], i}, i = Position[d, _?(# > 0&)] // Flatten; PalindromeQ[d[[First[i];;Last[i]]]]]; q[1] = True; Select[Range[1000], q]
%Y A373085 Cf. A000142, A007623, A046807, A294168.
%K A373085 nonn,base
%O A373085 1,2
%A A373085 _Amiram Eldar_, May 23 2024