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brilliant n.  and  Poulet n.

First numbers that belong to both sets:

341, 1387, 2047, 2701, 31621, 42799, 49141, 49981, 60701, 83333, 88357, 90751, 104653, 129889, 130561, 150851, 162193, 164737, 219781, 226801, 241001, 249841, 282133, 1194649, 1678541, 2035153, 2134277, 2163001, 2284453, 2304167, 2491637, 2977217, 3059101, 3090091, 3125281, 3235699, 3345773, 3363121, 3375041, 3400013, 3542533, 3568661, 3898129, 3985921, 4082653, 4101637, 4151869, 4181921, 4314967, 4360621, 4415251, 4469471, 4513841, 4567837, 4863127, 5044033, 5095177, 5173169, 5176153, 5351537, 5423713, 5489641, 5524693, 5575501, 5766001, 5919187, 6027193, 6122551, 6226193, 6233977, 6236257, 6278533, 6368689, 6539527, 6658669, 6749021, 6779137, 6787327, 6912079, 6952037, 6955541, 6973063, 7232321, 7295851, 7306261, 7306561, 7725901, 7820201, 8036033, 8095447, 8231653, 8650951, 8725753, 8745277, 8902741, 9006401, 9037729, 9040013, 9056501, 9131401, 9273547, 9371251, 9480461, 9533701, 9591661, 9729301, 9863461, 10031653, 10084177, 10386241, 10402237, 10403641, 10425511, 10505701, 10610063, 10712857, 10974881, 11335501, 11367137, 11541307, 11585293, 12032021, 12273769, 12322133, 12327121, 12659989, 12711007, 13073941, 13295281, 13338371, 13694761, 13747361, 13773061, 13856417, 13991647, 13996951, 14026897, 14179537, 14282143, 14324473, 14589901, 14794081, 14899751, 15139199, 15188557, 15268501, 15525241, 15583153, 15621409, 15757741, 15976747, 15978007, 16070429, 16324001, 16349477, 16717061, 16853077, 16973393, 17116837, 17327773, 17375249, 17405537, 17870561, 18073817, 18443701, 18535177, 18653353, 18736381, 18985627, 19020191, 19328653, 19404139, 19985269, 20081953, 20117467, 20234341, 20261251, 21306157, 21355951, 21397381, 21654533, 21907009, 22137809, 22369621, 22591301, 22669501, 23247901, 23261713, 23315977, 23464033, 23963869, 24904153, 25276421, 25457833, 25696133, 25909453, 26470501, 26977001, 27108397, 27331921, 27409541, 27509653, 27664033, 27798461, 27808463, 28011001, 28325881, 28406953, 29581501, 30058381, 30529693, 30881551, 31040833, 32091781, 32701297, 34890481, 35851037, 36307981, 37962541, 38118763, 38439523, 40361197, 42485119, 42702661, 43363601, 45219329, 46094401, 46325029, 46517857, 47253781, 47918581, 48191653, 48448661, 49075417, 52181407, 53154337, 53283169, 53675623, 55318957, 55610837, 63001801, 63781901, 64605041, 128027831, 129461617, 133427449, 155840777, 158397247, 158496911, 158895281, 166082309, 166444181, 168566501, 177951973, 183554407, 186846301, 187667969, 193330237, 206028271, 207132481, 213035761, 214858717, 217472501, 217517653, 220261861, 220531501, 224582221, 224957893, 227132641, 227518271, 232794253, 234870301, 235742513, 237791143, 238001653, 242239321, 242729401, 242860069, 246099317, 246282511.

Some larger examples :