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admirable n.  and  gapful n.

First numbers that belong to both sets:

120, 140, 1204, 1484, 1638, 2002, 2480, 2828, 4095, 6435, 7425, 8415, 8925, 10444, 10724, 12124, 12404, 12964, 14084, 14644, 15764, 16256, 16604, 17164, 18004, 18284, 18844, 19124, 20482, 21028, 21308, 22708, 22988, 24384, 24668, 25508, 26348, 27188, 27748, 28588, 28868, 29428, 29708, 30240, 30555, 30592, 31815, 32445, 32760, 40640, 43065, 45356, 46035, 56896, 62992, 67952, 78975, 81081, 83312, 89408, 100324, 100604, 101164, 101444, 102004, 102844, 103455, 104524, 104656, 106204, 106484, 107044, 107324, 107884, 108164, 109844, 110404, 112084, 112364, 114044, 114604, 115724, 116284, 118804, 119084, 119536, 119644, 119924, 122164, 122444, 123844, 124496, 124964, 125524, 125804, 126364, 126644, 128324, 128884, 129195, 130004, 130564, 130844, 131684, 132244, 132524, 133924, 134204, 134416, 134764, 137284, 138124, 138176, 138404, 139244, 139376, 139804, 140084, 140644, 143164, 144284, 146524, 147644, 148484, 149044, 149324, 151004, 151564, 152404, 153524, 154084, 154256, 155764, 156044, 157444, 158284, 159124, 159404, 160804, 161924, 162764, 163604, 164176, 165284, 165844, 166684, 169204, 169484, 170044, 171164, 171724, 172004, 172564, 172844, 173684, 175364, 177044, 177604, 177884, 178444, 181244, 182655, 183484, 183764, 186284, 186844, 187684, 188524, 189364, 190204, 190484, 191044, 191324, 191565, 192164, 192724, 195524, 196364, 197204, 198884, 198896, 200228, 201908, 204988, 205268, 205828, 207508, 208628, 209468, 211148, 211708, 212548, 213388, 215068, 215348, 216748, 219548, 221228, 222628, 224308, 226268, 226828, 227108, 228508, 228788, 229348, 230188, 230468, 232148, 232708, 236068, 236908, 238028, 238588, 240268, 241948, 243068, 243892, 244468, 244748, 245308, 246988, 247268, 248108, 250348, 250628, 251188, 252028, 252308, 253148, 254012, 254548, 256228, 256508, 257068, 258188, 258748, 259868, 260708, 261261, 261548, 262388, 262948, 263788, 264068, 264908, 265748, 266308, 266588, 267428, 268828, 269668, 270508, 272188, 273868, 274148, 274708, 275828, 276388, 277228, 278068, 278348, 281708, 282548, 283108, 283948, 284228, 285068, 285908, 287588, 288428, 288988, 289268, 290948, 294028, 294868, 296828, 297668, 298228, 299348.

Some larger examples :