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gapful n.  and  zygodrome

First numbers that belong to both sets:

1100, 1111, 1155, 1188, 2200, 2222, 3300, 3333, 4400, 4444, 4455, 5500, 5555, 6600, 6666, 7700, 7777, 8800, 8888, 9900, 9999, 11000, 11100, 11144, 22000, 22200, 22288, 22333, 33000, 33300, 33355, 44000, 44333, 44400, 44666, 55000, 55500, 66000, 66600, 66999, 77000, 77700, 88000, 88666, 88800, 99000, 99900, 110000, 110011, 110055, 110088, 111000, 111100, 111111, 111555, 111888, 112200, 112211, 113300, 113311, 113344, 113355, 114400, 114411, 115500, 115511, 116600, 116611, 116655, 117700, 117711, 118800, 118811, 118833, 119900, 119911, 119955, 119988, 220000, 220022, 220077, 221100, 221122, 222000, 222200, 222222, 222777, 223300, 223322, 223344, 223366, 224400, 224422, 225500, 225522, 226600, 226611, 226622, 226688, 227700, 227722, 228800, 228822, 229900, 229922, 229944, 229999, 330000, 330033, 330077, 331111, 332255, 333000, 333300, 333333, 333777, 336600, 336633, 338844, 339900, 339933, 339955, 440000, 440044, 440055, 441144, 442200, 442244, 443344, 444000, 444400, 444444, 444555, 445544, 446600, 446644, 446688, 447744, 448800, 448844, 449944, 449955, 550000, 550044, 550055, 551100, 551155, 552200, 552255, 552299, 553300, 553355, 554400, 554433, 554455, 555000, 555444, 555500, 555555, 556600, 556655, 557700, 557755, 558800, 558855, 558888, 559900, 559955, 660000, 660066, 662222, 663300, 663366, 666000, 666600, 666666, 669900, 669955, 669966, 770000, 770044, 770077, 777000, 777444, 777700, 777777, 880000, 880088, 882288, 883344, 884400, 884488, 886688, 888000, 888800, 888888, 990000, 990099, 993333, 993377, 998844, 999000, 999900, 999999, 1100000, 1100011, 1100055, 1100088, 1100333, 1110000, 1111000, 1111100, 1111155, 1112200, 1112244, 1113300, 1113333, 1114400, 1114455, 1114488, 1115500, 1116600, 1117700, 1117755, 1118800, 1119900, 1119944, 1119977, 1122000, 1122200, 1122255, 1122555, 1133000, 1133300, 1133355, 1133388, 1144000, 1144400, 1144444, 1144455, 1155000, 1155500, 1155555, 1155599, 1155888, 1166000, 1166600, 1166633, 1166655, 1166688, 1166999, 1177000, 1177700, 1177755, 1177777, 1188000, 1188555, 1188800, 1188855, 1199000, 1199900, 1199955, 1199988, 2200000, 2200022, 2200666, 2211000, 2211100, 2211111, 2211144, 2211188, 2211888, 2220000, 2221100, 2221133, 2222000, 2222200, 2222299, 2223300, 2224400.

Some larger examples :