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heptagonal n.  and  odious n.

First numbers that belong to both sets:

1, 7, 55, 81, 112, 148, 286, 342, 403, 1177, 1288, 1404, 1651, 1918, 2356, 2512, 2839, 3367, 3553, 3744, 3940, 4141, 4558, 5929, 6175, 6943, 7480, 7756, 8037, 8910, 9517, 10791, 12145, 12496, 13213, 13579, 14326, 14707, 15484, 16687, 17935, 18361, 19228, 21022, 22896, 24850, 25351, 25857, 26368, 26884, 27931, 28998, 29539, 31192, 31753, 33466, 34633, 35820, 36421, 38254, 38875, 40132, 40768, 42055, 43362, 46036, 46717, 47403, 49491, 50908, 51624, 52345, 53071, 53802, 54538, 55279, 56776, 60606, 62173, 62964, 63760, 64561, 65367, 66994, 68641, 69472, 71995, 73702, 74563, 77176, 79834, 81631, 82537, 83448, 84364, 86211, 88078, 90916, 93799, 95746, 97713, 98704, 100701, 101707, 102718, 103734, 104755, 107848, 108889, 114169, 116316, 117397, 118483, 119574, 122877, 126225, 128482, 130759, 131905, 134212, 135373, 136539, 140067, 141253, 144841, 147258, 152152, 153388, 154629, 157126, 160909, 166023, 168610, 171217, 177822, 180499, 181845, 188650, 190026, 191407, 192793, 194184, 199798, 201214, 208369, 209815, 214183, 217120, 221563, 223054, 226051, 227557, 232105, 238239, 239785, 241336, 246019, 249166, 250747, 253924, 265201, 266832, 268468, 275062, 285103, 290191, 291897, 295324, 297045, 298771, 307476, 309232, 310993, 312759, 316306, 319873, 325261, 327067, 330694, 332515, 334341, 336172, 338008, 341695, 343546, 345402, 351000, 352876, 356643, 358534, 360430, 362331, 364237, 368064, 371911, 373842, 379665, 381616, 385533, 391446, 393427, 395413, 397404, 401401, 405418, 413512, 415548, 419635, 423742, 425803, 427869, 429940, 432016, 436183, 440370, 442471, 448804, 450925, 453051, 461605, 463756, 468073, 470239, 472410, 474586, 483340, 489958, 492174, 496621, 501088, 514609, 516880, 523723, 526014, 535228, 542191, 546858, 549199, 553896, 556252, 558613, 570493, 572884, 577681, 580087, 582498, 587335, 592192, 594628, 597069, 599515, 611820, 614296, 616777, 634284, 636805, 639331, 644398, 646939, 652036, 654592, 659719, 664866, 670033, 675220, 677821, 680427, 685654, 688275, 693532, 696168, 698809, 706762, 714760, 722803, 725494, 733597, 739024, 744471, 749938.

Some larger examples :