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Cunningham n.  and  happy n.

First numbers that belong to both sets:

7, 10, 28, 31, 82, 129, 226, 362, 440, 899, 901, 1088, 1090, 1330, 1332, 1599, 1727, 2026, 2115, 2188, 2602, 2745, 3026, 3135, 3137, 3365, 3376, 3968, 3970, 5477, 6083, 6560, 7397, 7745, 8099, 10815, 11880, 12099, 12101, 12770, 12995, 13457, 13825, 14399, 15375, 15377, 15624, 15877, 16899, 17162, 17423, 17690, 18226, 20448, 23103, 25280, 27226, 28560, 28562, 29242, 29792, 30977, 31330, 33123, 34224, 35343, 35722, 35938, 36482, 38417, 39203, 39602, 40400, 41615, 41617, 42435, 42437, 42850, 43682, 47088, 47523, 48401, 49285, 50175, 51075, 53360, 55226, 55695, 57120, 62002, 63002, 63503, 64515, 66050, 68122, 69170, 73442, 75624, 78401, 78960, 79508, 80088, 81226, 82945, 84101, 87024, 88210, 90001, 91124, 93026, 95482, 101125, 101762, 102399, 103822, 103824, 104328, 105624, 106277, 108899, 110223, 110591, 112895, 114920, 117648, 118335, 118337, 119024, 121105, 121800, 125001, 130320, 131770, 132652, 134688, 142130, 143642, 145160, 148876, 151322, 152099, 152101, 156024, 156817, 163217, 166376, 168099, 168101, 168922, 169743, 169745, 171397, 173055, 173057, 176401, 178085, 178928, 184040, 185192, 185194, 188355, 189226, 190095, 190968, 191845, 192720, 197137, 199808, 205208, 206117, 207937, 210682, 211599, 214368, 214370, 217157, 219960, 225626, 226577, 226982, 227530, 235226, 238143, 238327, 239122, 243048, 244035, 244037, 246015, 247008, 247010, 248005, 250048, 255026, 257050, 262143, 262145, 265226, 266255, 270401, 271440, 274577, 277728, 278783, 281962, 287495, 287497, 288370, 289443, 292682, 293763, 294850, 298115, 306917, 310248, 312480, 314720, 316970, 322625, 328510, 329477, 330626, 339888, 341057, 348101, 354024, 363610, 370882, 380690, 386885, 389018, 393130, 394383, 395640, 396901, 400690, 403224, 405768, 405770, 408322, 419903, 423802, 430335, 434280, 440897, 444890, 465123, 467857, 476099, 476101, 480250, 483024, 483026, 489999, 492805, 495617, 501265, 502680, 502682, 504099, 505520, 506943, 512001, 522730, 525626, 527077, 529985, 537825, 541697, 543170, 546120, 546122, 549082, 552050, 562501, 568515, 570024, 574565, 577599, 589825.

Some larger examples :