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constructible polygon  and  evil n.

First numbers that belong to both sets:

3, 5, 6, 10, 12, 15, 17, 20, 24, 30, 34, 40, 48, 51, 60, 68, 80, 85, 96, 102, 120, 136, 160, 170, 192, 204, 240, 255, 257, 272, 320, 340, 384, 408, 480, 510, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 3855, 4080, 4112, 4352, 4369, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 13107, 15360, 15420, 16320, 16448, 17408, 17476, 20480, 20560, 21760, 21845, 24576, 24672, 26112, 26214, 30720, 30840, 32640, 32896, 34816, 34952, 40960, 41120, 43520, 43690, 49152, 49344, 52224, 52428, 61440, 61680, 65280, 65535, 65537, 65792, 69632, 69904, 81920, 82240, 87040, 87380, 98304, 98688, 104448, 104856, 122880, 123360, 130560, 131070, 131074, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 196611, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 262148, 263168, 278528, 279616, 327680, 327685, 328960, 348160, 349520, 393216, 393222, 394752, 417792, 419424, 491520, 493440, 522240, 524280, 524296, 526336, 557056, 559232, 655360, 655370, 657920, 696320, 699040, 786432, 786444, 789504, 835584, 838848, 983040, 983055, 986880, 1044480, 1048560, 1048592, 1052672, 1114112, 1114129, 1118464, 1310720, 1310740, 1315840, 1392640, 1398080, 1572864, 1572888, 1579008, 1671168, 1677696, 1966080, 1966110, 1973760, 2088960, 2097120, 2097184, 2105344, 2228224, 2228258, 2236928, 2621440, 2621480, 2631680, 2785280, 2796160, 3145728, 3145776, 3158016, 3342336, 3342387, 3355392, 3932160, 3932220, 3947520, 4177920, 4194240, 4194368, 4210688, 4456448, 4456516, 4473856, 5242880, 5242960, 5263360, 5570560, 5570645, 5592320, 6291456, 6291552, 6316032, 6684672, 6684774, 6710784, 7864320, 7864440, 7895040, 8355840, 8388480, 8388736, 8421376, 8912896, 8913032, 8947712, 10485760, 10485920, 10526720, 11141120, 11141290, 11184640, 12582912, 12583104, 12632064, 13369344, 13369548, 13421568, 15728640, 15728880, 15790080, 16711680, 16711935, 16776960, 16777472, 16842752, 16843009, 17825792.

Some larger examples :