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cyclic n.  and  gapful n.

First numbers that belong to both sets:

143, 187, 341, 451, 473, 561, 583, 671, 781, 1001, 1037, 1111, 1159, 1207, 1245, 1313, 1335, 1349, 1441, 1547, 1551, 1605, 1661, 1695, 1717, 1729, 1919, 1963, 1991, 2059, 2093, 2323, 2929, 3115, 3131, 3333, 3367, 3395, 3605, 3737, 3745, 3815, 4141, 4277, 4343, 4747, 5151, 5353, 5369, 5959, 6097, 6161, 6305, 6695, 6767, 6955, 6969, 7171, 7373, 7777, 7979, 8099, 8245, 8383, 8787, 8989, 9191, 9215, 9785, 9797, 10005, 10021, 10047, 10153, 10155, 10217, 10245, 10279, 10351, 10387, 10461, 10543, 10673, 10681, 10727, 10785, 10897, 11039, 11063, 11145, 11231, 11237, 11341, 11407, 11419, 11451, 11453, 11561, 11577, 11595, 11609, 11671, 11747, 11917, 11955, 11985, 11989, 12001, 12135, 12179, 12233, 12331, 12405, 12551, 12559, 12585, 12623, 12661, 12749, 12767, 12855, 12881, 12883, 12937, 12945, 12991, 13107, 13129, 13211, 13273, 13277, 13305, 13319, 13321, 13403, 13515, 13541, 13651, 13663, 13699, 13787, 13793, 13871, 13889, 13935, 13957, 13981, 14183, 14201, 14205, 14269, 14295, 14311, 14443, 14467, 14573, 14655, 14745, 14807, 14833, 14839, 14861, 14963, 14971, 14977, 15045, 15191, 15195, 15285, 15301, 15353, 15409, 15411, 15487, 15521, 15599, 15613, 15735, 15743, 15961, 15997, 16003, 16133, 16167, 16169, 16181, 16291, 16455, 16507, 16511, 16523, 16545, 16549, 16621, 16635, 16739, 16783, 16841, 16847, 17017, 17119, 17171, 17173, 17281, 17309, 17445, 17501, 17527, 17563, 17611, 17693, 17697, 17805, 17831, 17867, 17879, 17895, 17941, 17953, 18037, 18161, 18255, 18285, 18343, 18345, 18381, 18435, 18449, 18547, 18829, 18863, 18885, 18887, 18931, 19019, 19057, 19123, 19151, 19155, 19245, 19253, 19335, 19371, 19397, 19399, 19567, 19589, 19591, 19605, 19643, 19785, 19811, 19903, 19907, 19921, 19969, 20033, 20039, 20263, 20329, 20723, 20953, 21199, 21643, 21779, 21873, 22069, 22333, 22359, 22649, 22793, 23483, 23519, 23713, 23809, 23943, 24173, 24403, 24679, 25093.

Some larger examples :