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congruent n.  and  hexagonal n.

First numbers that belong to both sets:

6, 15, 28, 45, 120, 190, 231, 276, 325, 496, 561, 630, 703, 861, 1326, 1431, 1653, 1770, 2016, 2145, 2278, 2415, 2556, 2701, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4560, 4950, 5151, 5356, 5565, 6441, 6670, 6903, 7140, 7381, 8128, 8385, 8646, 8911, 9180, 9453, 9730, 10011, 10296, 10878, 11175, 11476, 11781, 13041, 13366, 13695, 14365, 15400, 15753, 16110, 16471, 17205, 17955, 18336, 19110, 19503, 19900, 20301, 21115, 21528, 21945, 22366, 22791, 23220, 23653, 24976, 25878, 26335, 27261, 28680, 29646, 30135, 31125, 32640, 33670, 34191, 34716, 35245, 36856, 37950, 38503, 39060, 39621, 41328, 41905, 42486, 43071, 44253, 44850, 46665, 47278, 47895, 49141, 51040, 52326, 52975, 53628, 54285, 56280, 57630, 58311, 58996, 59685, 61776, 63190, 63903, 65341, 66066, 67528, 68265, 69006, 69751, 71253, 73536, 75078, 75855, 76636, 77421, 78210, 79800, 81406, 82215, 83028, 83845, 87990, 88831, 90525, 93961, 94830, 95703, 97461, 99235, 101926, 102831, 103740, 104653, 106491, 107416, 109278, 110215, 111156, 112101, 116886, 117855, 119805, 124750, 125751, 127765, 130816, 131841, 132870, 133903, 134940, 135981, 139128, 140185, 141246, 142311, 143380, 144453, 145530, 147696, 149878, 150975, 153181, 154290, 156520, 158766, 159895, 162165, 163306, 165600, 167910, 169071, 170236, 171405, 173755, 174936, 176121, 177310, 178503, 179700, 180901, 184528, 186966, 188191, 189420, 190653, 193131, 195625, 196878, 198135, 200661, 207046, 208335, 209628, 210925, 214840, 217470, 218791, 220116, 221445, 225456, 228150, 229503, 230860, 232221, 237705, 239086, 240471, 243253, 248865, 250278, 251695, 253116, 254541, 255970, 258840, 260281, 261726, 263175, 264628, 266085, 273430, 274911, 277885, 285390, 286903, 289941, 291466, 296065, 297606, 299151, 300700, 302253, 303810, 305371, 306936, 310078, 311655, 313236, 314821, 319600, 321201, 322806, 324415, 327645, 334153, 335790, 337431, 340725, 345696, 349030, 350703, 352380, 354061, 359128, 362526, 364231, 365940, 367653, 372816, 376278, 378015, 381501, 386760, 390286, 392055, 395605.

Some larger examples :