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equidigital n.  and  palindromic n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 11, 101, 111, 121, 131, 141, 151, 161, 181, 191, 313, 353, 373, 383, 686, 727, 757, 787, 797, 919, 929, 2662, 3773, 3993, 10001, 10301, 10401, 10501, 10601, 10801, 10901, 11111, 11311, 11411, 11611, 11711, 11911, 12021, 12321, 12421, 12521, 12621, 12721, 12821, 12921, 13031, 13331, 13531, 13631, 13831, 13931, 14041, 14141, 14341, 14441, 14741, 14941, 15451, 15551, 15751, 16061, 16161, 16361, 16561, 16661, 16961, 17071, 17371, 17471, 17571, 17671, 17971, 18181, 18481, 18781, 18881, 19191, 19391, 19491, 19691, 19791, 19891, 19991, 20402, 25952, 30103, 30203, 30403, 30603, 30703, 30803, 31013, 31513, 31913, 32323, 32423, 32723, 33433, 33533, 33733, 34043, 34343, 34543, 34643, 34843, 34943, 35053, 35153, 35353, 35753, 36163, 36263, 36463, 36563, 36763, 36963, 37273, 37573, 38083, 38183, 38783, 38983, 39293, 39493, 40704, 44544, 44944, 48384, 52625, 57875, 62426, 65856, 70207, 70507, 70607, 71317, 71917, 72227, 72727, 73037, 73237, 73637, 74047, 74747, 75557, 76367, 76667, 77377, 77477, 77977, 78487, 78587, 78787, 78887, 79397, 79697, 79997, 90209, 90709, 91019, 93139, 93239, 93639, 93739, 94049, 94349, 94649, 94849, 94949, 95959, 96269, 96469, 96769, 97379, 97579, 97879, 98389, 98689, 100001, 103301, 104401, 106601, 107701, 112211, 119911, 121121, 134431, 147741, 255552, 343343, 350053, 376673, 389983, 497794, 592295, 1000001, 1001001, 1003001, 1004001, 1005001, 1006001, 1008001, 1012101, 1013101, 1015101, 1016101, 1017101, 1018101, 1019101, 1021201, 1022201, 1024201, 1025201, 1026201, 1027201, 1028201, 1029201, 1032301, 1033301, 1035301, 1036301, 1038301, 1039301, 1041401, 1042401, 1043401, 1044401, 1045401, 1046401, 1047401, 1049401, 1052501, 1053501, 1054501, 1055501, 1058501, 1061601, 1062601, 1065601, 1066601, 1067601, 1068601, 1069601, 1073701, 1074701, 1076701, 1077701, 1079701, 1081801, 1082801, 1084801, 1085801, 1086801, 1091901, 1092901, 1093901, 1095901, 1096901, 1098901, 1099901, 1100011, 1101011, 1104011, 1106011, 1107011, 1109011, 1111111, 1114111, 1116111, 1117111, 1118111, 1120211, 1123211, 1125211, 1126211.

Some larger examples :