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Cunningham n.  and  cyclic n.

First numbers that belong to both sets:

3, 5, 7, 15, 17, 31, 33, 35, 37, 65, 101, 127, 143, 145, 197, 215, 217, 255, 257, 323, 401, 485, 511, 577, 677, 785, 899, 901, 1001, 1157, 1295, 1297, 1601, 1727, 1729, 1763, 1765, 1937, 2047, 2049, 2117, 2501, 2703, 2743, 2917, 3137, 3365, 3599, 3601, 3845, 4097, 4355, 4357, 4899, 5183, 5477, 5777, 5833, 6083, 6085, 6401, 7055, 7057, 7395, 7397, 7743, 7745, 7777, 7999, 8099, 8101, 8191, 8465, 8837, 9215, 9217, 9605, 10001, 10403, 10649, 10817, 11237, 11663, 11665, 12101, 12545, 12995, 12997, 13457, 13823, 14401, 14885, 15377, 15877, 16385, 16901, 17423, 17957, 18497, 19043, 19045, 19601, 20165, 20737, 21317, 21905, 21953, 22499, 22501, 23717, 24337, 24965, 25601, 26243, 26897, 27001, 27557, 28901, 30977, 31683, 31685, 32399, 32401, 32767, 33857, 34597, 35345, 36101, 36863, 37637, 38417, 39203, 39305, 40001, 41617, 42437, 43265, 44099, 44101, 45797, 46657, 48401, 49283, 49285, 50177, 51077, 51983, 52901, 54755, 54757, 54871, 55697, 56645, 57599, 57601, 59537, 60517, 62499, 62501, 63505, 64001, 64517, 65535, 65537, 66563, 66565, 67601, 68645, 69695, 69697, 70755, 70757, 71823, 72899, 72901, 73985, 74087, 74089, 75077, 76177, 78401, 79523, 80657, 81797, 82945, 84101, 85265, 86435, 86437, 87617, 90001, 91205, 92417, 93637, 94865, 96101, 97337, 97343, 97345, 98597, 99857, 100001, 101123, 102399, 102401, 103685, 104977, 106277, 107585, 108899, 108901, 110591, 111557, 112895, 112897, 114245, 115601, 116965, 118337, 119715, 119717, 121103, 121105, 122501, 125317, 126737, 128165, 129601, 131045, 131071, 131073, 132497, 133955, 133957, 136901, 138383, 138385, 139877, 141377, 142883, 144401, 147457, 148997, 150545, 152099, 152101, 153663, 155237, 156815, 156817, 157463, 158405, 160001, 161603, 163217, 164837, 166463, 168101, 169745, 171395, 171397, 173057, 176399, 176401, 178085, 179777, 181477, 183185, 184901, 186623, 188357, 190097, 191843, 193601, 195113, 197135, 197137, 198915, 198917, 200703.

Some larger examples :