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cyclic n.  and  pentagonal n.

First numbers that belong to both sets:

1, 5, 35, 51, 145, 247, 287, 1001, 1247, 1335, 1717, 2147, 2501, 3151, 4187, 5017, 5551, 6305, 6501, 7957, 8177, 10795, 11051, 11837, 13207, 14065, 15251, 16801, 17767, 20827, 25285, 26467, 26867, 31901, 33227, 35497, 36895, 37367, 38801, 43265, 44807, 45327, 46905, 47437, 49051, 49595, 51245, 54055, 56357, 58115, 63551, 65417, 67947, 68587, 71177, 73151, 76501, 81317, 84847, 96647, 105205, 106001, 108407, 112477, 114955, 118301, 122551, 128627, 133057, 146797, 147737, 151527, 155365, 159251, 167167, 174251, 179401, 182527, 187797, 192067, 199655, 205165, 208507, 209627, 213005, 223301, 226787, 232657, 236215, 237407, 241001, 250717, 255647, 261877, 265651, 272001, 277135, 281017, 282317, 296815, 298151, 302177, 308947, 313045, 314417, 318551, 325501, 329707, 331115, 335357, 342487, 346801, 352595, 364327, 365807, 371757, 388367, 396037, 402227, 406901, 414751, 421085, 425867, 427467, 433897, 438751, 445265, 458437, 460097, 465095, 480251, 485357, 493927, 499105, 506051, 514801, 520087, 541501, 563347, 565187, 587501, 593147, 602617, 610247, 625651, 631477, 649117, 655051, 663005, 679057, 681077, 713805, 722107, 730457, 736751, 747301, 753667, 762197, 772927, 805567, 807767, 832165, 834401, 841127, 859195, 861467, 868301, 879751, 886657, 895907, 907537, 914551, 916895, 926301, 935755, 942877, 945257, 952415, 974051, 981317, 993487, 1003277, 1010651, 1023001, 1030447, 1052947, 1090987, 1101245, 1121905, 1132307, 1134915, 1145377, 1153255, 1155887, 1163801, 1177051, 1187705, 1195727, 1209157, 1217251, 1241695, 1252637, 1285751, 1294097, 1308067, 1327751, 1341901, 1350427, 1361837, 1387685, 1396355, 1410865, 1419607, 1422527, 1431305, 1457801, 1466687, 1490515, 1493507, 1517551, 1526617, 1553977, 1563151, 1566215, 1590835, 1600117, 1603217, 1640651, 1665847, 1688051, 1713607, 1716815, 1726457, 1729677, 1781605, 1791427, 1794707, 1804565, 1830985, 1834301, 1844267, 1860937, 1901251, 1911397, 1914785, 1924967, 1941997, 1993537, 1996997, 2007395, 2035255, 2049257, 2066827, 2080937, 2109301, 2119987, 2134277, 2152207, 2163001, 2206447, 2239315, 2254001, 2265047, 2283517, 2294635, 2298347, 2309501, 2328151, 2339377, 2354387, 2384551, 2418715, 2430157, 2445455, 2491637, 2511007, 2522665, 2526557, 2538251, 2557801.

Some larger examples :