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centered pentagonal n.  and  cyclic n.

First numbers that belong to both sets:

1, 31, 51, 141, 181, 331, 391, 601, 681, 951, 1051, 1381, 1501, 1891, 2031, 2481, 2641, 3151, 3331, 3901, 4101, 4951, 5641, 5881, 6631, 6891, 7981, 8851, 9151, 10081, 10401, 11391, 11731, 12781, 13141, 14251, 14631, 17431, 17851, 19141, 20931, 21391, 23281, 24751, 27301, 28891, 29431, 31081, 33351, 33931, 35701, 36301, 38131, 38751, 41281, 43231, 43891, 45901, 46581, 48651, 49351, 51481, 52201, 54391, 58141, 60451, 61231, 63601, 64401, 66831, 67651, 70141, 70981, 77001, 77881, 80551, 81451, 84181, 87891, 88831, 91681, 92641, 95551, 100501, 103531, 104551, 107641, 111831, 112891, 116101, 117181, 120451, 124881, 126001, 129391, 130531, 133981, 139831, 143401, 144601, 148231, 153141, 154381, 158131, 159391, 163201, 169651, 173581, 174901, 180231, 185641, 189751, 191131, 200931, 202351, 206641, 208081, 212431, 218301, 219781, 224251, 225751, 230281, 231801, 236391, 237931, 242581, 244141, 248851, 250431, 256801, 261631, 263251, 268141, 274731, 276391, 281401, 283081, 288151, 294981, 296701, 301891, 303631, 308881, 317731, 323101, 324901, 330331, 339481, 345031, 346891, 352501, 354381, 360051, 361951, 367681, 369601, 375391, 377331, 383181, 385141, 391051, 393031, 399001, 407031, 409051, 415141, 417181, 423331, 433681, 439951, 442051, 456891, 459031, 465481, 467641, 474151, 476331, 482901, 485101, 491731, 493951, 500641, 502881, 511891, 518701, 520981, 527851, 539401, 546391, 555781, 565251, 567631, 574801, 584431, 594141, 596581, 603931, 613801, 616281, 626251, 633781, 636301, 643891, 656641, 664351, 666931, 674701, 677301, 687751, 695641, 698281, 706231, 708891, 716901, 719581, 727651, 730351, 738481, 741201, 749391, 752131, 760381, 763141, 771451, 785401, 793831, 796651, 805141, 807981, 819391, 828001, 830881, 839551, 842451, 854101, 862891, 865831, 874681, 886551, 889531, 901501, 910531, 913551, 925681, 934831, 937891, 947101, 959451, 962551, 971881, 975001, 984391, 987531, 1000141, 1009651, 1012831, 1022401, 1038451, 1048141, 1051381, 1061131, 1077481, 1087351, 1090651, 1100581, 1117231, 1127281, 1130641, 1140751, 1144131, 1157701, 1167931, 1171351, 1181641, 1185081, 1195431, 1209301, 1212781, 1223251.

Some larger examples :