In base 7, these positive numbers and their squares are palindromic.
1, 2, 4, 8, 32, 40, 50, 57, 64, 200, 344, 400, 1376, 1432, 2402, 2451, 2500, 2752, 2801, 9608, 9804, 16808, 17200, 19216, 19608, 67232, 68408, 117650, 117993, 118336, 120100, 120443, 134464, 134807, 136914, 470600, 471972, 823544, 826288, 840400, 843144, 941200
1
Written in base 7, the numbers are 1, 2, 4, 11, 44, 55, 101, 111, 121, 404, 1001, 1111, 4004, 4114, 10001,…. The squares are 1, 4, 22, 121, 2662, 4444, 10201, 12321, 14641, 224422, 1002001, 1234321, 22044022, 23300332, 100020001,….
T. D. Noe, Plot of 257 terms
T. D. Noe, Table of 257 terms
Eric W. Weisstein, MathWorld: Palindromic Number
(Mma) makePalindrome[n_Integer, b_Integer, del_] := Module[{c = IntegerDigits[n, b], d}, d = If[del, Join[c, Reverse[Most[c]]], Join[c, Reverse[c]]]; FromDigits[d]]; palindromeQ[n_, b_] := Module[{d = IntegerDigits[n, b]}, d == Reverse[d]]; b = 7; t = {}; Do[Do[Do[d = makePalindrome[i, b, j]; e = FromDigits[IntegerDigits[d], b]; If[palindromeQ[e^2, b], AppendTo[t, e]], {i, b^(n - 1), b^n - 1}], {j, {True, False}}], {n, Floor[0.5 + 10*Log[3]/Log[b]]}]
nonn,base
T. D. Noe, Aug 26 2016