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cake n.  and  odious n.

First numbers that belong to both sets:

1, 2, 4, 8, 26, 42, 64, 93, 176, 299, 988, 1160, 1562, 2048, 2325, 2952, 3683, 4526, 5489, 6580, 7807, 8474, 9920, 11522, 16262, 17344, 18473, 23479, 27776, 29317, 30914, 34280, 47972, 52463, 62269, 64898, 67600, 70376, 76154, 79158, 85401, 88642, 95368, 98855, 106082, 113653, 129859, 147537, 156948, 166751, 187565, 193026, 198592, 210043, 215930, 228032, 267034, 273938, 280960, 288101, 302744, 310248, 317875, 333502, 349633, 366276, 374792, 383439, 392218, 401130, 410176, 428674, 438128, 477334, 487488, 497785, 529544, 597058, 608840, 670080, 695682, 708724, 721928, 748826, 776384, 804610, 924354, 940108, 956040, 1021568, 1038405, 1055426, 1072632, 1161472, 1179809, 1217060, 1353602, 1373904, 1415115, 1436026, 1478464, 1521730, 1588199, 1656576, 1679797, 1774851, 1799162, 1848448, 1873425, 1924052, 1949704, 2001690, 2028026, 2054592, 2081389, 2108418, 2163176, 2218874, 2247078, 2304201, 2362284, 2451226, 2542373, 2604376, 2635752, 2862468, 2929551, 3032128, 3066845, 3137072, 3172584, 3280726, 3317312, 3391298, 3466376, 3504327, 3581058, 3619840, 3658901, 3737864, 3777768, 3817955, 3858426, 3981553, 4023170, 4065076, 4149759, 4192538, 4278976, 4500251, 4545402, 4682665, 4729026, 4775692, 4822664, 4869943, 4917530, 5013632, 5062149, 5110978, 5160120, 5209576, 5259347, 5309434, 5461601, 5512962, 5564644, 5616648, 5668975, 5774602, 5935490, 5989776, 6099339, 6154618, 6210230, 6379074, 6436028, 6550951, 6725888, 6844226, 7024323, 7146126, 7269329, 7519967, 7841282, 7906624, 7972328, 8038395, 8238784, 8306313, 8374210, 8442476, 8580119, 8649498, 8719250, 8859877, 9002008, 9437505, 9585732, 9810970, 9886826, 10116738, 10271976, 10428794, 10507798, 10587200, 10667001, 10747202, 10990215, 11072026, 11319893, 11403330, 11487176, 11571432, 11741178, 11826670, 11998897, 12260360, 12348351, 12436762, 12525594, 12614848, 12885152, 13159290, 13437289, 13530818, 13624780, 14392184, 14490088, 14588435, 14687226, 14786462, 15289352, 15391279, 15596490, 15699776, 15803517, 15907714, 16223051, 16435574, 16649945, 17084263, 17304226, 17414912, 17526069, 18088954, 18202958, 18317440, 18432401, 18663764, 18897055, 19014426, 19369453, 19488770, 19728872, 19849659, 20337737, 20460994, 20584748, 20709000, 20833751, 21084754, 21592792, 21721064, 21849843, 22108926, 22370049, 22633220, 22765576, 22898447, 23031834, 23165738, 23435101, 23570562, 23843048, 24255702, 24533433, 24673090, 24953992, 25237018.

Some larger examples :