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centered triangular n.  and  odious n.

First numbers that belong to both sets:

1, 4, 19, 31, 64, 109, 199, 274, 316, 361, 409, 460, 631, 829, 901, 976, 1054, 1135, 1219, 1306, 1585, 1684, 1891, 1999, 2341, 2461, 2839, 3385, 3529, 3676, 4135, 4294, 4456, 4621, 4789, 4960, 5134, 5311, 5860, 6049, 6241, 6436, 7039, 7246, 7456, 7885, 8551, 9244, 9481, 9721, 10711, 11485, 11749, 12559, 13969, 14260, 15454, 15760, 16069, 16381, 16696, 17014, 17335, 17659, 17986, 18316, 18649, 19324, 19666, 20359, 20710, 21064, 21421, 21781, 22144, 22510, 23251, 23626, 24769, 25156, 26335, 27541, 27949, 28774, 29191, 30460, 31321, 31756, 32194, 32635, 33079, 34429, 34885, 37210, 37684, 38641, 39124, 39610, 42085, 42589, 44119, 46201, 47794, 49414, 52735, 53299, 54436, 55009, 55585, 56164, 57331, 59701, 61510, 62731, 63346, 63964, 64585, 65836, 67099, 67735, 68374, 69016, 70309, 71614, 72271, 72931, 73594, 74260, 74929, 76276, 77635, 78319, 79006, 80389, 81085, 81784, 82486, 83899, 86761, 88210, 88939, 90406, 91144, 91885, 93376, 94879, 97156, 97921, 98689, 99460, 101791, 103360, 104941, 105736, 107335, 111385, 112204, 114679, 115510, 116344, 117181, 119710, 120559, 121411, 122266, 123124, 124849, 125716, 127459, 130096, 130981, 131869, 134551, 135451, 139081, 141835, 143686, 146485, 149311, 151210, 152164, 153121, 154081, 155044, 156979, 161869, 166834, 169849, 173911, 176989, 178021, 183226, 184276, 185329, 186385, 187444, 190639, 192784, 194941, 197110, 198199, 201484, 204796, 205906, 207019, 208135, 209254, 210376, 212629, 213760, 214894, 216031, 218314, 221761, 222916, 225235, 228736, 229909, 234631, 237010, 243010, 246646, 249085, 251536, 252766, 255235, 258961, 260209, 263971, 265231, 266494, 269029, 271576, 272854, 274135, 275419, 277996, 279289, 280585, 283186, 287110, 289741, 292384, 293710, 295039, 296371, 297706, 299044, 303076, 308494, 311221, 315334, 319474, 323641, 325036, 327835, 330646, 336304, 337726, 339151, 340579, 342010, 355024, 359416, 360886, 363835, 365314, 366796, 368281, 369769, 372754, 374251, 375751, 377254, 378760, 381781, 384814, 387859, 389386, 393985.

Some larger examples :