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centered triangular n.  and  equidigital n.

First numbers that belong to both sets:

1, 10, 19, 31, 64, 109, 166, 199, 235, 361, 409, 571, 631, 829, 1135, 1219, 1306, 1489, 1585, 1891, 1999, 2341, 2971, 3385, 3529, 4135, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 10966, 11485, 11749, 12016, 12286, 12559, 13114, 13681, 13969, 14851, 15454, 16069, 16381, 17335, 17659, 18985, 19666, 20011, 20359, 21421, 21781, 22144, 23251, 24385, 25939, 27541, 29191, 29611, 31321, 34429, 34885, 35344, 36739, 37210, 40099, 40591, 42589, 44119, 46201, 46729, 48871, 50509, 51061, 53299, 55009, 57331, 62119, 62731, 69661, 70309, 72271, 72931, 74929, 89671, 98689, 101011, 101791, 102574, 104149, 104941, 106534, 107335, 108139, 110569, 111385, 113851, 114679, 120559, 121411, 124849, 127459, 128335, 129214, 130981, 131869, 134551, 135451, 136354, 138169, 139081, 140914, 142759, 143686, 145549, 146485, 149311, 153121, 154081, 156979, 157951, 161869, 164839, 166834, 169849, 172891, 173911, 176989, 178021, 182179, 185329, 186385, 188506, 189571, 190639, 193861, 198199, 203689, 207019, 208135, 211501, 213760, 216031, 217171, 226399, 228736, 229909, 239401, 244219, 245431, 253999, 255235, 260209, 263971, 265231, 269029, 275419, 284491, 289741, 292384, 295039, 300385, 301729, 312589, 318091, 322249, 323641, 334885, 339151, 340579, 344881, 346321, 352111, 356485, 363835, 380269, 381781, 386335, 387859, 393985, 398611, 403264, 406381, 412651, 417385, 436591, 438211, 443089, 451279, 457885, 462871, 464539, 469561, 471241, 476299, 477991, 489919, 491635, 510709, 519499, 524809, 533719, 539101, 540901, 570109, 577531, 598189, 601984, 605791, 613441, 621139, 623071, 626944, 652411, 660349, 684451, 686479, 692581, 700759, 711049, 717259, 736051, 742369, 770059, 776521, 778681, 811441, 813649, 847129, 865261, 867541, 885889, 913771, 961201, 968419, 1000009, 1004914, 1007371, 1009831, 1012294, 1019701, 1024654, 1027135, 1029619, 1032106, 1037089, 1047091, 1049599, 1054624, 1057141, 1059661, 1067239, 1069771, 1079929, 1087579, 1090135, 1097821, 1100389, 1108111, 1121041, 1126234, 1128835, 1131439, 1134046, 1139269, 1149751, 1160281, 1170859, 1173511, 1176166, 1181485, 1184149, 1186816, 1194835, 1202881, 1210954, 1213651.

Some larger examples :