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cyclic n.  and  triangular n.

First numbers that belong to both sets:

1, 3, 15, 91, 435, 561, 595, 703, 1891, 2701, 3655, 4465, 5151, 7021, 8911, 9453, 10011, 10153, 10585, 12403, 15051, 15753, 16471, 18721, 22791, 23005, 23653, 26335, 35245, 38503, 39621, 41041, 44551, 49141, 50403, 50721, 52003, 55945, 59685, 62481, 68635, 69751, 77421, 77815, 79003, 88831, 93961, 99235, 103285, 104653, 110215, 115921, 121771, 125751, 129795, 132355, 133903, 138601, 142845, 144453, 146611, 153181, 157641, 159895, 174345, 188191, 193753, 201295, 205761, 218791, 226801, 229503, 241165, 243253, 248865, 249571, 257403, 266815, 269011, 277885, 286903, 293761, 303031, 314821, 322003, 334153, 337431, 344035, 364231, 374545, 385003, 392055, 400065, 403651, 406351, 417241, 425503, 435711, 439453, 458403, 459361, 473851, 485605, 497503, 502503, 506521, 521731, 534061, 542361, 543403, 568711, 571915, 597871, 621055, 624403, 637885, 648091, 651511, 665281, 680361, 688551, 693253, 697971, 718201, 721801, 736291, 761995, 765703, 795691, 826255, 836571, 837865, 841753, 873181, 889111, 890445, 901153, 905185, 915981, 921403, 926841, 928203, 954271, 966745, 970921, 982101, 983503, 993345, 994755, 1004653, 1027461, 1033203, 1038961, 1056331, 1073845, 1109305, 1127251, 1134771, 1145341, 1177345, 1181953, 1195831, 1220703, 1276003, 1295245, 1301691, 1314631, 1329265, 1334161, 1348903, 1353835, 1373653, 1393615, 1395285, 1408681, 1474903, 1490401, 1495585, 1511191, 1516411, 1537381, 1551441, 1553203, 1558495, 1601155, 1622701, 1688203, 1755001, 1771903, 1794565, 1800253, 1807851, 1809753, 1830741, 1840321, 1869211, 1884711, 1886653, 1892485, 1910035, 1963171, 1987021, 2035153, 2133145, 2158003, 2166321, 2208151, 2233441, 2252503, 2284453, 2303731, 2318781, 2329561, 2362051, 2381653, 2388205, 2407915, 2460871, 2467531, 2487565, 2521135, 2548153, 2575315, 2595781, 2602621, 2623195, 2630071, 2639253, 2678455, 2685403, 2706301, 2722611, 2724945, 2741311, 2769481, 2790703, 2826253, 2847691, 2854855, 2857245, 2895621, 2912491, 2963395, 2970703, 3029491, 3051685, 3059101, 3081403, 3101295, 3111265, 3118753, 3141271, 3161355, 3168903, 3171421, 3178981, 3209311, 3239785, 3262735, 3270403, 3293461, 3301165, 3303735, 3332071, 3363121, 3386503, 3417805, 3425653, 3449251, 3488761, 3531153, 3584503, 3616705, 3627471, 3649051, 3681541, 3706003, 3736011, 3738745, 3771631.

Some larger examples :