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centered triangular n.  and  congruent n.

First numbers that belong to both sets:

31, 46, 85, 109, 136, 166, 199, 316, 514, 631, 694, 760, 829, 901, 976, 1054, 1135, 1396, 1684, 1999, 2110, 2341, 2461, 2584, 2710, 2839, 3676, 4135, 4294, 4621, 4789, 4960, 5134, 5311, 7039, 7246, 7669, 7885, 8326, 8551, 9244, 10711, 10966, 11224, 11485, 11749, 12016, 12286, 12559, 13114, 13396, 14260, 14554, 15151, 15454, 15760, 16069, 16381, 16696, 17014, 17335, 17986, 19324, 20359, 20710, 21421, 21781, 22510, 22879, 24004, 24385, 26335, 26734, 27541, 27949, 28774, 29191, 30460, 32194, 33079, 33526, 33976, 34429, 34885, 35806, 36271, 37684, 39124, 40591, 41086, 41584, 42085, 42589, 43096, 43606, 44119, 47260, 48871, 49414, 50509, 51061, 52174, 52735, 57919, 58510, 59104, 59701, 60301, 61510, 62119, 63964, 64585, 67735, 68374, 69016, 69661, 70309, 71614, 72271, 74260, 76276, 76954, 78319, 79006, 79696, 80389, 81085, 81784, 82486, 83191, 87484, 88939, 89671, 90406, 91885, 92629, 94126, 94879, 101791, 102574, 103360, 104149, 104941, 106534, 107335, 109756, 113026, 114679, 115510, 116344, 117181, 118021, 118864, 119710, 120559, 123124, 125716, 128335, 129214, 130981, 131869, 132760, 133654, 134551, 139996, 142759, 143686, 145549, 146485, 147424, 148366, 149311, 150259, 157951, 158926, 160885, 161869, 163846, 164839, 167836, 168841, 169849, 173911, 174934, 175960, 176989, 178021, 179056, 180094, 181135, 184276, 187444, 190639, 191710, 193861, 194941, 196024, 197110, 198199, 200386, 204796, 208135, 209254, 211501, 212629, 214894, 216031, 222916, 226399, 227566, 228736, 229909, 231085, 233446, 234631, 238204, 245431, 246646, 247864, 249085, 250309, 251536, 252766, 253999, 257716, 261460, 265231, 266494, 267760, 269029, 270301, 271576, 272854, 274135, 280585, 281884, 283186, 285799, 287110, 288424, 289741, 291061, 293710, 295039, 307135, 308494, 309856, 311221, 312589, 315334, 316711, 320860, 323641, 329239, 330646, 332056, 333469, 334885, 337726, 339151, 343444, 344881, 347764, 352111, 353566, 355024, 356485, 357949, 359416, 360886, 362359, 371260, 375751, 377254, 380269, 381781, 383296, 384814, 386335, 400159.

Some larger examples :