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Cunningham n.  and  gapful n.

First numbers that belong to both sets:

120, 143, 170, 195, 242, 360, 440, 1001, 1090, 1155, 1330, 1332, 1368, 1370, 1443, 1520, 1680, 1729, 1850, 1935, 2400, 2600, 3480, 3720, 4095, 4224, 4760, 5775, 6240, 7777, 8099, 8282, 9024, 9215, 9408, 9999, 10200, 10610, 10815, 11235, 11237, 11448, 11450, 11880, 12168, 12320, 12770, 13455, 13690, 13923, 14160, 14640, 15130, 15375, 15624, 16128, 16130, 16640, 17160, 17690, 17955, 18495, 18770, 19320, 19684, 19880, 21024, 22200, 22800, 23408, 24024, 24388, 25280, 25920, 26243, 26999, 27888, 28225, 28560, 29240, 30275, 30975, 32040, 32760, 35343, 36480, 39303, 39305, 39600, 40400, 42435, 43680, 44520, 47088, 47960, 48840, 49728, 54288, 59050, 60515, 63000, 63505, 65024, 67080, 75075, 77840, 78960, 82368, 83520, 87024, 100001, 100490, 101760, 103040, 103824, 104328, 104330, 106275, 106930, 108240, 109560, 110890, 111555, 112224, 112897, 113570, 114920, 116280, 117648, 117650, 118335, 118337, 119715, 120410, 121800, 123200, 123903, 124610, 126735, 127450, 128880, 129599, 130320, 131770, 132495, 132650, 133224, 134690, 136160, 137640, 139130, 139875, 140607, 141375, 142128, 142130, 142883, 143640, 145160, 146690, 148878, 148995, 149770, 151320, 152880, 154450, 155235, 157608, 157610, 159200, 159999, 160800, 161050, 161052, 161603, 162410, 163215, 163217, 164024, 164835, 165650, 167280, 168920, 170568, 170570, 173055, 173890, 175560, 177240, 178930, 179775, 182330, 183183, 184040, 185760, 187488, 187490, 188355, 189224, 190095, 190970, 192720, 194480, 196250, 198915, 198917, 199810, 201600, 203400, 205380, 207024, 210680, 212520, 214368, 219025, 219960, 221840, 224675, 225624, 226575, 226980, 227528, 229440, 231360, 232323, 232325, 235224, 239120, 241080, 249000, 251000, 253008, 255024, 259080, 261120, 265224, 265226, 267288, 268325, 269360, 271440, 274575, 275626, 276675, 278783, 279840, 281960, 283025, 284088, 286224, 290520, 292680, 297024, 299208, 300764, 303600, 306915, 312480, 313599, 315843, 320355, 323760, 326040, 337560, 343397, 346920, 352835, 355215, 358800, 361200, 368448, 372099, 373320, 383160, 389016, 389375, 394383, 395640, 398160, 405225.

Some larger examples :