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Cunningham n.  and  economical n.

First numbers that belong to both sets:

3, 5, 7, 10, 15, 17, 31, 35, 37, 101, 122, 127, 129, 145, 197, 215, 217, 224, 257, 401, 485, 511, 577, 677, 1025, 1088, 1157, 1226, 1295, 1297, 1445, 1522, 1601, 1682, 1763, 1765, 2047, 2049, 2117, 2303, 2305, 2501, 2705, 2917, 3137, 3365, 3599, 3845, 3968, 4357, 4625, 5183, 5477, 5831, 6399, 6723, 7057, 8101, 8191, 8837, 10001, 10202, 10405, 10649, 10817, 11237, 11665, 12101, 12997, 13457, 13690, 13823, 13825, 13925, 14162, 14401, 14642, 15375, 15377, 15875, 15877, 16128, 16640, 16901, 17162, 17577, 17957, 18497, 19322, 19882, 20737, 21317, 21951, 21953, 22501, 23105, 23717, 24337, 24965, 25280, 25601, 25920, 28901, 29585, 30977, 31685, 32401, 33125, 33857, 35345, 37248, 37637, 38417, 39205, 40805, 41617, 42437, 44101, 49285, 50177, 52901, 55697, 57122, 57601, 58563, 59535, 60517, 62501, 65024, 65537, 66048, 67601, 69697, 72901, 78401, 80657, 90001, 93637, 98597, 100001, 101125, 101760, 102401, 103040, 103042, 103685, 104977, 105626, 106277, 107585, 108242, 108899, 108901, 110225, 110593, 111557, 112226, 112895, 114245, 115601, 116963, 118335, 118337, 119026, 119717, 121103, 121105, 121802, 122501, 123905, 124999, 125315, 126737, 128165, 129599, 131045, 131071, 131073, 132497, 133955, 133957, 135425, 136901, 137642, 138383, 139875, 140609, 140626, 141377, 142885, 143642, 144401, 146688, 147457, 148224, 148878, 148997, 150545, 152101, 152882, 153665, 156815, 156817, 157463, 159202, 160001, 161605, 163217, 164026, 164837, 167282, 168101, 171395, 172226, 173057, 176399, 176401, 178085, 180626, 183185, 184901, 186623, 186625, 189226, 190097, 193601, 194482, 195113, 195365, 196250, 197135, 197137, 198026, 199808, 206117, 207937, 209765, 213443, 213445, 215297, 215999, 216001, 217157, 219023, 219025, 220901, 224677, 228485, 232325, 234257, 236195, 237168, 238145, 238327, 240101, 242065, 244037, 246017, 248003, 250001, 250048, 254017, 258065, 260099, 260101, 263168, 266257, 270401, 272483, 272485, 274624, 276677, 279935, 285157, 287297, 287497, 295937, 309137, 311365, 313601.

Some larger examples :