Numbers whose reciprocals have a palindromic repeating part in base 6.
5, 10, 15, 30, 45, 60, 90, 180, 270, 344, 360, 540, 1080, 1620, 2064, 2160, 2997, 3240, 3483, 6480, 9720, 10449, 12384, 12960, 17982, 19440, 20898, 38880, 58320, 62694, 74304, 77760, 107892, 116640, 125388, 233280, 349920, 376164, 445824, 466560, 647352, 699840, 752328, 1399680
1
The repeating part of the reciprcals in base 6 is longer than 1 digit for numbers of the form (344, 2997, 3483, 10449) * 6^(i-1) for i = 1, 2, 3,… sorted. The other numbers appear to be (5, 10, 15, 45) * 6^(i-1) for i = 1, 2, 3,… sorted. For example the fraction 1/344 in base 6 is 0.000343343….
T. D. Noe, Plot of 44 terms
Wikipedia, Repeating decimal
(Mma) Select[Range[6^6], (r = RealDigits[1/#, 6]; ! IntegerQ[r[[1, 1]]] && r[[1, 1]] == Reverse[r[[1, 1]]]) &]
nonn,base
T. D. Noe, Jun 03 2016