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octagonal n.  and  odious n.

First numbers that belong to both sets:

1, 8, 21, 133, 176, 280, 341, 481, 560, 936, 1160, 1281, 1408, 1825, 1976, 2133, 2465, 2821, 3008, 3400, 3816, 4033, 4256, 4485, 4720, 5208, 5461, 5985, 6256, 6816, 7105, 7400, 7701, 8008, 8321, 8965, 10680, 11408, 12936, 13736, 14145, 14981, 15841, 16280, 17176, 18565, 19040, 19521, 20501, 22533, 23056, 25208, 26320, 28033, 28616, 29205, 29800, 31621, 34133, 36080, 39445, 40136, 43681, 44408, 45141, 46625, 48133, 48896, 50440, 52008, 52801, 54405, 55216, 56033, 58520, 61061, 64533, 68101, 69921, 70840, 71765, 73633, 75525, 77441, 80360, 81345, 82336, 83333, 84336, 86360, 87381, 89441, 90480, 92576, 93633, 94696, 95765, 97921, 99008, 100101, 101200, 102305, 105656, 107920, 109061, 110208, 111361, 112520, 113685, 116033, 118405, 120801, 122008, 123221, 126896, 128133, 129376, 131880, 133141, 134408, 136960, 143445, 144760, 150080, 152776, 156865, 161008, 162401, 169456, 170885, 172320, 179585, 182533, 184016, 190008, 193040, 199176, 202280, 203841, 206981, 208560, 210145, 211736, 214936, 218160, 219781, 221408, 223041, 226325, 227976, 229633, 231296, 234640, 236321, 238008, 239701, 244816, 249985, 253461, 256961, 258720, 260485, 264033, 265816, 269400, 271201, 273008, 274821, 283976, 285825, 287680, 291408, 293281, 295160, 297045, 304645, 306560, 308481, 312341, 314280, 318176, 328021, 332001, 334000, 342056, 346120, 348161, 350208, 356385, 358456, 360533, 364705, 368901, 371008, 379496, 381633, 383776, 396760, 403333, 405536, 409960, 416641, 421125, 425633, 432440, 434721, 443905, 446216, 448533, 453185, 457861, 464920, 467285, 469656, 472033, 476805, 479200, 481601, 484008, 491265, 501025, 505941, 508408, 510881, 518336, 523336, 528360, 530881, 538480, 546133, 553840, 559008, 561601, 577285, 582561, 590520, 593185, 612008, 614721, 628376, 631125, 636641, 642181, 656136, 661760, 678776, 681633, 684496, 690240, 698901, 704705, 707616, 710533, 716385, 722261, 728161, 731120, 740033, 746005, 749000, 755008, 764065, 767096, 770133, 776225, 782341, 785408, 791560, 797736, 800833, 807045, 810160, 813281, 816408, 822680, 832133, 835296, 838465, 841640, 844821.

Some larger examples :