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modest n.  and  palindromic n.

First numbers that belong to both sets:

111, 222, 333, 444, 515, 545, 555, 666, 777, 818, 888, 999, 1111, 1881, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 10101, 11111, 18081, 20202, 22222, 26962, 30303, 33333, 40404, 44444, 50505, 51515, 54545, 55555, 60606, 66666, 70707, 77777, 80108, 80808, 81818, 84148, 88888, 90909, 91819, 98189, 99999, 111111, 180081, 198891, 222222, 333333, 418814, 444444, 555555, 666666, 777777, 888888, 999999, 1001001, 1010101, 1111111, 1386831, 1400041, 1800081, 1980891, 2002002, 2020202, 2121212, 2222222, 2727272, 2800082, 3003003, 3030303, 3141413, 3333333, 4004004, 4040404, 4242424, 4444444, 4801084, 5005005, 5050505, 5151515, 5252525, 5454545, 5555555, 5757575, 6006006, 6060606, 6282826, 6363636, 6666666, 7007007, 7070707, 7425247, 7777777, 8008008, 8080808, 8121218, 8181818, 8484848, 8888888, 9009009, 9018109, 9090909, 9999999, 10899801, 11011011, 11111111, 12000021, 15400451, 16800861, 17200271, 18000081, 18188181, 19000091, 19800891, 19988991, 22022022, 22222222, 24000042, 31800813, 33033033, 33333333, 36000063, 36222263, 43566534, 44044044, 44444444, 48000084, 55055055, 55555555, 66066066, 66666666, 77077077, 77777777, 88088088, 88888888, 99099099, 99999999, 100010001, 101010101, 101101101, 108909801, 111111111, 112838211, 121121121, 123303321, 131131131, 132000231, 141141141, 141404141, 151151151, 153000351, 155404551, 161161161, 169404961, 171171171, 180000081, 181181181, 181808181, 184000481, 189202981, 191191191, 198000891, 198959891, 199808991, 200020002, 202020202, 202202202, 207818702, 212121212, 212212212, 220414022, 222222222, 232232232, 242242242, 246606642, 252252252, 259202952, 262262262, 264000462, 272272272, 272727272, 282282282, 282808282, 292292292, 296808692, 300030003, 303030303, 303303303, 306727603, 307131703, 313313313, 323323323, 333333333, 343343343, 349808943, 353353353, 363363363, 369909963, 373373373, 383383383, 393393393, 396000693, 400040004, 404040404, 404404404, 405606504, 405636504, 414414414, 418000814, 424242424, 424424424, 434434434, 435606534, 440828044, 444444444, 454454454, 464464464, 474474474, 484484484, 494494494, 500050005, 504545405, 505050505, 505505505, 512525215, 515151515, 515515515, 525030525, 525252525, 525525525, 535505535, 535535535, 545454545, 545545545, 555555555, 563585365, 565565565, 575575575, 575757575, 585585585, 595595595, 597040795, 600060006, 603454306, 606060606, 606606606, 616616616, 626626626, 636363636, 636636636, 646646646, 656656656, 666666666, 676676676, 686686686, 696696696, 700070007, 702363207, 707070707, 707707707, 713707317, 717717717, 727727727, 737737737, 747747747, 748000847, 757757757, 767767767, 777777777, 787787787, 797797797, 800080008, 801272108, 808080808.

Some larger examples :