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Duffinian n.  and  gapful n.

First numbers that belong to both sets:

100, 121, 143, 187, 225, 242, 253, 275, 341, 363, 385, 400, 451, 484, 583, 671, 781, 800, 900, 1037, 1053, 1111, 1159, 1183, 1207, 1250, 1296, 1313, 1331, 1349, 1443, 1539, 1573, 1575, 1600, 1661, 1703, 1771, 1875, 1919, 1936, 1963, 1991, 2025, 2059, 2075, 2125, 2187, 2304, 2323, 2331, 2425, 2500, 2525, 2575, 2675, 2775, 2783, 2825, 2875, 2929, 2975, 2997, 3131, 3441, 3535, 3600, 3605, 3737, 3745, 3751, 3872, 3955, 4141, 4343, 4747, 4773, 4949, 4961, 5184, 5335, 5353, 5555, 5665, 5885, 5959, 6161, 6305, 6565, 6767, 6771, 6955, 6993, 7171, 7275, 7373, 7725, 7777, 7979, 8245, 8383, 8755, 8989, 8991, 9025, 9216, 9797, 10000, 10021, 10153, 10217, 10279, 10351, 10387, 10469, 10543, 10571, 10673, 10681, 10727, 10803, 10816, 10849, 10897, 10901, 11063, 11231, 11237, 11323, 11325, 11341, 11407, 11419, 11453, 11561, 11609, 11671, 11713, 11747, 11775, 11843, 11891, 11917, 11973, 11989, 12001, 12100, 12179, 12221, 12225, 12233, 12257, 12331, 12369, 12427, 12493, 12551, 12559, 12623, 12661, 12749, 12767, 12800, 12881, 12883, 12937, 12991, 13013, 13125, 13129, 13211, 13273, 13277, 13319, 13321, 13403, 13431, 13447, 13456, 13509, 13575, 13651, 13663, 13699, 13725, 13787, 13793, 13871, 13889, 13957, 13981, 14053, 14175, 14201, 14269, 14297, 14311, 14313, 14400, 14443, 14450, 14459, 14467, 14475, 14531, 14573, 14641, 14839, 14861, 14963, 14971, 14977, 15075, 15093, 15191, 15223, 15301, 15317, 15353, 15409, 15483, 15487, 15521, 15599, 15613, 15631, 15789, 15825, 15979, 15997, 16003, 16133, 16181, 16291, 16337, 16393, 16425, 16507, 16511, 16523, 16549, 16621, 16725, 16739, 16783, 16841, 16847, 16900, 16913, 16951, 17119, 17173, 17281, 17303, 17357, 17499, 17527, 17611, 17693, 17823, 17831, 17867, 17879, 17941, 17953, 18037, 18050, 18075, 18083, 18161, 18207, 18225, 18259, 18271, 18343, 18377, 18449, 18491, 18496, 18547.

Some larger examples :