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centered triangular n.  and  prime n.

First numbers that belong to both sets:

19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589, 44119, 48871, 51061, 53299, 55009, 57331, 62119, 62731, 69661, 70309, 72271, 72931, 74929, 89671, 98689, 104149, 108139, 110569, 114679, 130981, 142759, 145549, 154081, 156979, 157951, 161869, 164839, 176989, 178021, 182179, 190639, 193861, 211501, 244219, 253999, 260209, 265231, 269029, 275419, 289741, 295039, 312589, 322249, 323641, 339151, 340579, 346321, 352111, 380269, 398611, 406381, 412651, 436591, 438211, 443089, 451279, 462871, 464539, 469561, 471241, 476299, 477991, 510709, 519499, 533719, 539101, 540901, 570109, 577531, 598189, 613441, 621139, 623071, 652411, 660349, 684451, 686479, 692581, 700759, 711049, 717259, 736051, 742369, 770059, 776521, 778681, 811441, 847129, 865261, 867541, 885889, 913771, 961201, 968419, 1019701, 1037089, 1049599, 1067239, 1079929, 1139269, 1173511, 1184149, 1202881, 1213651, 1216351, 1227181, 1246249, 1268221, 1270981, 1282051, 1293169, 1301539, 1315549, 1335289, 1358029, 1427401, 1442071, 1501501, 1513531, 1525609, 1546861, 1562131, 1571329, 1574401, 1586719, 1611499, 1620841, 1633339, 1658479, 1671121, 1674289, 1699741, 1760959, 1803469, 1813351, 1816651, 1843159, 1869859, 1879921, 1910269, 1934041, 1947691, 2006239, 2058619, 2072701, 2076229, 2086831, 2118799, 2129509, 2143831, 2147419, 2172619, 2201599, 2216161, 2234431, 2245429, 2304541, 2308261, 2383291, 2394649, 2398441, 2413639, 2425069, 2455681, 2471059, 2501959, 2552581, 2568259, 2580049, 2611621, 2615581, 2627479, 2659339, 2679349, 2695411, 2707489, 2711521, 2727679, 2756071, 2788699, 2891899, 2904409, 2921131, 2988499, 3009709, 3022471, 3039529, 3147229, 3199591, 3217141, 3283381, 3287821, 3305611, 3354781, 3426949, 3486151, 3518239, 3541249, 3559711, 3578221, 3610729, 3742231, 3818431, 3823219, 3842401, 3939031, 3953629, 3992689, 4017199, 4036861, 4111021, 4130911, 4150849, 4155841, 4215979, 4236121, 4251259, 4276549, 4291759, 4393849, 4414411, 4419559, 4481569, 4523149, 4538791, 4622671, 4643761, 4670191, 4814209, 4835731, 4841119, 4878919, 4987729, 5053591, 5081161, 5125429, 5253769, 5281879, 5349649, 5417851, 5549779.

Some larger examples :