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centered triangular n.  and  twin primes

First numbers that belong to both sets:

19, 31, 109, 199, 571, 829, 1489, 1999, 2341, 2971, 3529, 4789, 9721, 10459, 10711, 13681, 16069, 17659, 20359, 27541, 31321, 48871, 51061, 57331, 72271, 89671, 104149, 110569, 142759, 145549, 154081, 164839, 182179, 193861, 211501, 244219, 260209, 295039, 322249, 340579, 352111, 380269, 398611, 451279, 464539, 510709, 570109, 577531, 598189, 613441, 660349, 770059, 865261, 1019701, 1037089, 1079929, 1216351, 1246249, 1335289, 1427401, 1442071, 1501501, 1513531, 1525609, 1562131, 1633339, 1699741, 1910269, 2072701, 2086831, 2129509, 2143831, 2201599, 2245429, 2413639, 2455681, 2471059, 2501959, 2627479, 2659339, 2695411, 2756071, 2921131, 2988499, 3199591, 3287821, 3486151, 3559711, 3742231, 3953629, 3992689, 4150849, 4155841, 4419559, 4481569, 4523149, 5081161, 5125429, 5253769, 5549779, 5665789, 6021019, 6069199, 6099409, 6123631, 6190489, 6269971, 6719359, 6744781, 6846949, 6872611, 7027591, 7099201, 7158061, 7289731, 7336099, 7496191, 7523041, 7766851, 8386291, 8607631, 9110641, 9162469, 9251659, 9499159, 10418791, 10426699, 11290189, 11331379, 12068599, 13351909, 13940029, 14040811, 14151169, 14224981, 14308249, 14973661, 15712399, 15751261, 15858379, 16142521, 16330051, 17345101, 17539471, 18380251, 18464359, 18622579, 18919729, 19391431, 19434601, 20098891, 20219869, 20529451, 20573869, 21302389, 21426931, 21894331, 22343611, 22855969, 23102551, 24138199, 24963481, 25110559, 25159681, 25258069, 25467781, 25566769, 25665949, 25752889, 25952161, 26516731, 26820319, 27074629, 27728551, 28520941, 28835761, 29046601, 29244961, 29364301, 29617039, 29777221, 30381751, 30422269, 30598159, 32566411, 32580391, 32692339, 33765421, 34150819, 34222429, 35422111, 36894961, 38049499, 38170471, 38580169, 38885059, 39252511, 40318561, 41999959, 42190669, 42381811, 43086361, 43812931, 44316991, 44905969, 45515359, 45697321, 46445491, 47773639, 48402121, 48743551, 49844191, 50608009, 51588541, 51941011, 53721361, 53991001, 54441901, 54640891, 55659559, 55879069, 56264251, 57279331, 57725119, 58490671, 59449981, 60683941, 61276909, 62917579, 63834079, 64167211, 66523411, 67264669, 67727521, 69840229, 70188661, 70414579, 70929379, 71094511, 71756959, 71902279, 72986401, 75710881, 76458691, 77081089, 79137649, 79159441, 83019241, 83399089, 87451291, 87657571, 88577311, 89015869, 89293411, 90128629, 93035251, 94101481, 95245489, 96372361, 96468571, 96877999, 97360789, 97457491, 99181939, 103621549, 103821121, 103920979, 104245849, 105324031, 105424609, 105625909, 106054309, 107573239, 108158359, 109282141, 109307749, 110514709, 113026621, 117099091, 119336221.

Some larger examples :