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cyclic n.  and  heptagonal n.

First numbers that belong to both sets:

1, 7, 235, 403, 469, 697, 1177, 1651, 2059, 2839, 3367, 3553, 4141, 5221, 6943, 9211, 9517, 10465, 12145, 13213, 13579, 14707, 16687, 19669, 21949, 23377, 25351, 27931, 29539, 31753, 34633, 37027, 42055, 44023, 44689, 46717, 53071, 55279, 59059, 61387, 62173, 64561, 68641, 74563, 78943, 81631, 82537, 89965, 92833, 93799, 96727, 101707, 108889, 114169, 117397, 118483, 121771, 130759, 135373, 141253, 144841, 164737, 171217, 176491, 180499, 187279, 191407, 192793, 196981, 204061, 209815, 214183, 221563, 227557, 244453, 246019, 258727, 265201, 270109, 278389, 298771, 303979, 310993, 319873, 327067, 341695, 349129, 354757, 364237, 371911, 377719, 393427, 395413, 401401, 417589, 425803, 436183, 442471, 461605, 476767, 487747, 496621, 503329, 514609, 521437, 523723, 530611, 542191, 549199, 551545, 558613, 577681, 580087, 587335, 606883, 609349, 629257, 646939, 659719, 667447, 670033, 677821, 690901, 701455, 722803, 730891, 733597, 763693, 766459, 774787, 797215, 800041, 822829, 831457, 843031, 857611, 866419, 878233, 893113, 902101, 938503, 941569, 950797, 966277, 978751, 988159, 1003939, 1016653, 1026241, 1042321, 1052029, 1065043, 1081423, 1091311, 1094617, 1121245, 1144807, 1161787, 1203049, 1213477, 1216963, 1245031, 1269853, 1287733, 1298521, 1342123, 1345789, 1390171, 1401379, 1420159, 1431487, 1446661, 1465741, 1477249, 1492663, 1523731, 1527637, 1539385, 1559065, 1570933, 1574899, 1586827, 1606807, 1622881, 1634989, 1671583, 1683871, 1704451, 1716859, 1754353, 1817743, 1822009, 1834837, 1856317, 1873591, 1921507, 1925893, 1939081, 1961161, 1978915, 2028151, 2032657, 2068885, 2082553, 2087119, 2123827, 2156209, 2179489, 2193517, 2212291, 2235871, 2250079, 2269093, 2292973, 2365363, 2370229, 2384857, 2424085, 2429011, 2483527, 2488513, 2503501, 2528581, 2543689, 2563903, 2589283, 2604571, 2609677, 2650705, 2666173, 2671339, 2686867, 2712847, 2733721, 2749429, 2775709, 2791537, 2812711, 2839291, 2855299, 2860645, 2925187, 2968615, 2984983, 3006877, 3034357, 3056431, 3073039, 3100819, 3117547, 3123133, 3139921, 3168001, 3184909, 3190555, 3207523, 3252991, 3258697, 3321793, 3327559, 3373867, 3397141, 3443929, 3461557, 3485131, 3514711, 3532519, 3538465, 3556333, 3586213, 3604201, 3610207, 3676603, 3682669, 3700897, 3731377, 3755851, 3774259, 3823567, 3829753.

Some larger examples :