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Cunningham n.  and  Duffinian n.

First numbers that belong to both sets:

8, 9, 35, 50, 63, 65, 129, 143, 215, 217, 242, 323, 325, 399, 485, 511, 513, 575, 785, 899, 901, 999, 1025, 1157, 1295, 1443, 1445, 1682, 1727, 1763, 1765, 1937, 2047, 2117, 2303, 2305, 2501, 2705, 2743, 2915, 3365, 3599, 3601, 3843, 4097, 4355, 4625, 4901, 5183, 5185, 5475, 5777, 5831, 5833, 6083, 6085, 6399, 6401, 7055, 7397, 7775, 7777, 7999, 8001, 8193, 8465, 9217, 9605, 10001, 10403, 10405, 10649, 10817, 11237, 11663, 11665, 12099, 12545, 12995, 12997, 13823, 13925, 15875, 16383, 17423, 17425, 17577, 18497, 19043, 19045, 19601, 20737, 21951, 21953, 22499, 23105, 23717, 24335, 24965, 26243, 26897, 27001, 27557, 28223, 29585, 30275, 30277, 31685, 32399, 32767, 33123, 33125, 34595, 34597, 36101, 36863, 36865, 37637, 38417, 39203, 39205, 39305, 39999, 40805, 44099, 45797, 46655, 46657, 47523, 47525, 48401, 49283, 49285, 50175, 51077, 51983, 51985, 53825, 54755, 54757, 54871, 57122, 57599, 58563, 58565, 59537, 60517, 61505, 63503, 63505, 64517, 66563, 66565, 69695, 70757, 71825, 72899, 73985, 74087, 74089, 75077, 76175, 76177, 77283, 79523, 79525, 81797, 82945, 84099, 84101, 85183, 85265, 86437, 87617, 88805, 92417, 93635, 94863, 94865, 96101, 97337, 97343, 99857, 100001, 101123, 104977, 107585, 108899, 108901, 110223, 110593, 111557, 112895, 112897, 114245, 116963, 118337, 119717, 121103, 121105, 122499, 123905, 124999, 125315, 125317, 128165, 129599, 129601, 132497, 133955, 133957, 135425, 136901, 138383, 139877, 140609, 141377, 142883, 142885, 144399, 144401, 147455, 152099, 152101, 153665, 155237, 157463, 158403, 158405, 161603, 161605, 163217, 166463, 166465, 168101, 169745, 171397, 173057, 175615, 176399, 178085, 179777, 181475, 181477, 183185, 186623, 186625, 188357, 191843, 191845, 195111, 195113, 195365, 198917, 200705, 202499, 202501, 206117, 207937, 209765, 211601, 213443, 215999, 216001, 219023, 219025, 220899, 222785, 226577, 228485, 230399, 230401, 232325, 234257, 236195, 236197, 238143, 238327, 242063, 242065.

Some larger examples :