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Cunningham n.  and  odious n.

First numbers that belong to both sets:

7, 8, 26, 28, 31, 35, 37, 50, 82, 122, 124, 127, 143, 145, 168, 217, 224, 242, 244, 290, 342, 362, 440, 511, 530, 575, 577, 626, 675, 677, 728, 842, 899, 901, 962, 1001, 1090, 1226, 1330, 1332, 1368, 1522, 1682, 1727, 1729, 1763, 1765, 1850, 2024, 2047, 2198, 2208, 2303, 2305, 2402, 2602, 2703, 2705, 2745, 2808, 2915, 2917, 3026, 3124, 3248, 3374, 3376, 3482, 3599, 3601, 3720, 3968, 4226, 4490, 4760, 4912, 5042, 5328, 5475, 5477, 5626, 5833, 5930, 6242, 6560, 6858, 6860, 6888, 7055, 7057, 7226, 7570, 7777, 7922, 7999, 8001, 8099, 8101, 8191, 8282, 8648, 8835, 8837, 9026, 9215, 9217, 9260, 9410, 9800, 10202, 10610, 10649, 10815, 10817, 11024, 11235, 11237, 11448, 11663, 11665, 11880, 12168, 12320, 12770, 12995, 12997, 13224, 13690, 13825, 14160, 14399, 14401, 14642, 14883, 14885, 15128, 15626, 15875, 15877, 16130, 16642, 16808, 17162, 17575, 17688, 18226, 18768, 19043, 19045, 19322, 19682, 19684, 19880, 20450, 21026, 21315, 21317, 21610, 21903, 21905, 21951, 21953, 22202, 22800, 23410, 24024, 24390, 24650, 25280, 25920, 26243, 26245, 26570, 27001, 27224, 27555, 27557, 27890, 28223, 28225, 28562, 29240, 29790, 29930, 30626, 31328, 32042, 32399, 32401, 32762, 32767, 33488, 34226, 34595, 34597, 34968, 35343, 35345, 35722, 35936, 36480, 36863, 36865, 37250, 38024, 38810, 39203, 39205, 39305, 39600, 40402, 41208, 42024, 42848, 43263, 43265, 43682, 44099, 44101, 44520, 44943, 44945, 45368, 45795, 45797, 46226, 46655, 46657, 47090, 47523, 47525, 47960, 48840, 49283, 49285, 49730, 50626, 50652, 51530, 51983, 51985, 52442, 52899, 52901, 53362, 54288, 54873, 55226, 56168, 56643, 56645, 57122, 57599, 57601, 58080, 59050, 59318, 59535, 59537, 60024, 60515, 60517, 61010, 62000, 63002, 63503, 63505, 64001, 64008, 65024, 66050, 67082, 68120, 68920, 69170, 70224, 70755, 70757, 71290, 72360.

Some larger examples :