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hexagonal n.  and  odious n.

First numbers that belong to both sets:

1, 28, 91, 276, 496, 861, 1225, 1431, 1540, 1653, 1770, 1891, 2850, 3003, 3160, 3655, 4186, 4371, 4560, 4753, 4950, 5151, 5356, 5565, 5995, 8128, 10296, 10585, 10878, 11175, 11476, 12090, 12403, 13366, 13695, 14365, 15051, 16110, 16836, 17205, 18721, 19110, 19900, 20301, 21115, 21528, 21945, 22791, 24531, 24976, 25425, 25878, 26335, 26796, 27730, 28203, 29161, 30135, 30628, 31125, 31626, 32131, 35778, 38503, 43071, 43660, 44850, 46056, 47278, 49141, 49770, 51040, 53628, 54285, 56953, 58996, 60378, 61776, 62481, 63190, 65341, 67528, 68265, 70500, 72010, 72771, 73536, 74305, 75078, 76636, 79003, 81406, 82215, 84666, 85491, 87990, 88831, 89676, 90525, 91378, 93096, 93961, 97461, 98346, 101025, 103740, 104653, 109278, 112101, 114003, 114960, 117855, 118828, 119805, 120786, 124750, 130816, 137026, 141246, 142311, 143380, 144453, 145530, 148785, 149878, 154290, 158766, 159895, 161028, 162165, 164451, 165600, 167910, 173755, 174936, 176121, 184528, 185745, 188191, 189420, 190653, 193131, 198135, 200661, 203203, 210925, 214840, 216153, 217470, 218791, 220116, 221445, 230860, 234955, 236328, 237705, 243253, 251695, 263175, 264628, 266085, 267546, 269011, 271953, 273430, 274911, 279378, 282376, 283881, 286903, 289941, 296065, 297606, 300700, 303810, 306936, 308505, 311655, 313236, 319600, 324415, 327645, 329266, 337431, 339076, 340725, 344035, 345696, 347361, 350703, 352380, 357435, 359128, 360825, 364231, 367653, 372816, 376278, 378015, 379756, 381501, 383250, 385003, 390286, 392055, 399171, 400960, 402753, 406351, 409965, 426426, 431985, 439453, 441330, 445096, 450775, 458403, 462241, 464166, 468028, 481671, 483636, 485605, 489555, 491536, 493521, 505515, 507528, 509545, 513591, 515620, 517653, 519690, 521731, 536130, 544446, 546535, 548628, 550725, 563391, 569778, 571915, 574056, 576201, 584821, 586986, 591328, 597871, 611065, 615495, 617716, 619941, 624403, 631126, 633375, 635628, 644680, 649230, 651511, 658378, 667590, 669903, 672220, 674541, 679195, 681528, 688551, 690900, 693253, 695610, 697971, 700336, 702705, 705078, 717003, 724206, 726615, 729028, 746031, 748476.

Some larger examples :