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centered heptagonal n.  and  cyclic n.

First numbers that belong to both sets:

1, 43, 71, 197, 463, 547, 953, 1471, 1933, 2101, 2647, 2843, 3473, 3697, 4411, 4663, 5461, 5741, 6623, 6931, 7897, 8233, 9283, 9647, 10781, 11173, 12391, 12811, 14113, 14561, 15947, 16423, 17893, 18397, 19951, 20483, 22121, 22681, 24403, 24991, 26797, 27413, 29303, 29947, 31921, 32593, 34651, 35351, 37493, 38221, 40447, 41203, 43513, 46691, 47503, 49981, 50821, 54251, 56897, 57793, 60523, 61447, 65213, 68111, 72073, 73081, 76147, 77183, 81397, 84631, 85723, 89041, 90161, 93563, 94711, 98197, 99373, 102943, 104147, 107801, 109033, 112771, 114031, 117853, 119141, 123047, 124363, 128353, 129697, 133771, 135143, 139301, 140701, 144943, 146371, 150697, 152153, 158047, 162541, 164053, 168631, 170171, 174833, 176401, 181147, 182743, 187573, 189197, 194111, 195763, 200761, 202441, 207523, 214397, 216133, 221383, 228481, 230273, 235691, 237511, 243013, 244861, 250447, 252323, 257993, 259897, 265651, 267583, 273421, 275381, 281303, 283291, 289297, 291313, 297403, 299447, 305621, 307693, 313951, 316051, 322393, 324521, 330947, 333103, 339613, 341797, 348391, 350603, 357281, 359521, 366283, 368551, 375397, 377693, 384623, 386947, 393961, 396313, 403411, 405791, 415381, 422647, 425083, 432433, 434897, 442331, 444823, 454861, 462463, 465011, 472697, 475273, 485647, 493501, 504071, 506731, 514753, 517441, 525547, 528263, 536453, 539197, 547471, 550243, 558601, 561401, 569843, 572671, 581197, 584053, 592663, 595547, 604241, 607153, 615931, 618871, 627733, 630701, 639647, 642643, 651673, 654697, 663811, 666863, 676061, 679141, 688423, 691531, 700897, 704033, 713483, 716647, 726181, 729373, 738991, 742211, 751913, 764947, 768223, 781397, 794683, 804721, 808081, 818203, 821591, 831797, 835213, 845503, 848947, 859321, 862793, 873251, 876751, 890821, 901447, 905003, 915713, 919297, 930091, 933703, 944581, 948221, 959183, 962851, 973897, 977593, 988723, 992447, 1003661, 1007413, 1018711, 1022491, 1033873, 1037681, 1049147, 1064533, 1068397, 1080031, 1083923, 1095641, 1099561, 1111363, 1115311, 1127197, 1131173, 1143143, 1147147, 1159201, 1163233, 1175371, 1179431, 1191653, 1195741, 1208047, 1212163, 1224553, 1228697, 1241171, 1257901, 1262101.

Some larger examples :