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%I #8 May 26 2024 16:11:04
%S 1,2,3,6,8,9,10,12,20,24,30,40,60,120,126,144,160,180,189,210,240,315,
%T 360,384,630,720,840,896,1008,1056,1120,1260,1680,2240,2520,4480,5040,
%U 5184,5760,6048,6300,6720,6912,8064,9072,9450,10080,12096,13440,14400,18144
%N Numbers k such that the factorial base representation of 1/k without the leading zeros is palindromic.
%C All the factorials (A000142) are terms, since the factorial base representation of 1/k! is k-1 0's followed by 1.
%C If k > 4 is composite then (k-1)!/k is a term.
%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 The first 10 terms are:
%e n a(n) 1/a(n) in factorial base
%e -- ---- ------------------------
%e 1 1 1.
%e 2 2 0.1
%e 3 3 0.02
%e 4 6 0.01
%e 5 8 0.003
%e 6 9 0.00232
%e 7 10 0.0022
%e 8 12 0.002
%e 9 20 0.0011
%e 10 24 0.001
%t 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 Cf. A000142, A007623, A046807, A294168.
%K nonn,base
%O 1,2
%A _Amiram Eldar_, May 23 2024