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cyclic n.  and  Poulet n.

First numbers that belong to both sets:

341, 561, 1105, 1387, 1729, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18721, 19951, 23001, 23377, 29341, 30121, 30889, 31417, 31609, 31621, 35333, 39865, 41041, 42799, 46657, 49141, 49981, 52633, 60701, 60787, 62745, 63973, 65077, 65281, 68101, 74665, 75361, 80581, 83333, 83665, 85489, 88357, 88561, 90751, 91001, 93961, 101101, 104653, 107185, 113201, 115921, 123251, 126217, 129889, 130561, 137149, 150851, 157641, 158369, 162193, 162401, 164737, 172081, 176149, 181901, 188057, 188461, 194221, 196021, 196093, 204001, 215749, 219781, 220729, 226801, 228241, 241001, 249841, 252601, 253241, 256999, 264773, 271951, 272251, 275887, 276013, 278545, 280601, 282133, 284581, 285541, 294271, 294409, 314821, 318361, 323713, 334153, 340561, 341497, 348161, 357761, 367081, 387731, 390937, 396271, 399001, 401401, 410041, 422659, 423793, 427233, 435671, 443719, 448921, 449065, 452051, 458989, 464185, 481573, 486737, 488881, 489997, 493697, 493885, 512461, 513629, 514447, 526593, 530881, 534061, 552721, 556169, 563473, 574561, 580337, 582289, 587861, 588745, 604117, 611701, 617093, 622909, 625921, 635401, 642001, 647089, 653333, 656601, 657901, 658801, 665281, 665333, 665401, 670033, 672487, 679729, 680627, 683761, 688213, 710533, 721801, 722201, 722261, 729061, 738541, 741751, 742813, 743665, 745889, 748657, 769567, 769757, 786961, 800605, 818201, 825265, 838201, 838861, 841681, 847261, 852841, 873181, 875161, 877099, 898705, 915981, 916327, 934021, 950797, 976873, 983401, 997633, 1004653, 1016801, 1018921, 1024651, 1033669, 1050985, 1052503, 1052929, 1053761, 1064053, 1073021, 1082401, 1082809, 1092547, 1093417, 1104349, 1106785, 1109461, 1128121, 1132657, 1139281, 1141141, 1145257, 1152271, 1157689, 1168513, 1193221, 1207361, 1251949, 1252697, 1277179, 1293337, 1302451, 1306801, 1325843, 1333333, 1357441, 1357621, 1373653, 1397419, 1398101, 1419607, 1433407, 1441091, 1457773, 1461241, 1463749, 1472065, 1472353, 1472505, 1493857, 1500661, 1507561, 1507963, 1509709, 1520905, 1530787, 1533601, 1533961, 1534541, 1537381, 1549411, 1569457, 1579249, 1584133, 1615681, 1620385, 1643665.

Some larger examples :