Numbers whose reciprocals have a palindromic repeating part in base 10.

3, 9, 15, 18, 30, 45, 90, 150, 180, 225, 300, 450, 900, 1125, 1500, 1800, 2250, 3000, 4500, 9000, 11250, 15000, 18000, 22500, 30000, 45000, 90000, 112500, 150000, 180000, 225000, 300000, 450000, 900000

1

Note that the palindromic part of these numbers is but a single digit: 1/3 = 0.3333…, 1/9 = 0.1111…, 1/15 = 0.06666…, 1/18 = 0.05555…, 1/45 = 0.02222…, 1/225 = 0.004444…, and 1/1125 = 0.0008888…. It is somewhat surprising that all these numbers have a periodic part of length 1. See S000899 for similar numbers in base 6, for which the palindromic repeating part has length greater than 1. It appears that 6 is the only base that has lengths greater than 1.

T. D. Noe, Plot of 34 terms

Wikipedia, Repeating decimal

(Mma) Select[Range[10000], (r = RealDigits[1/#, 10]; ! IntegerQ[r[[1,1]]] && r[[1,1]] == Reverse[r[[1,1]]]) &]

Cf. A008776, A042950, A026395, S000898-S000903.

nonn,base,nice

T. D. Noe, Jun 01 2016