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amenable n.  and  cake n.

First numbers that belong to both sets:

1, 8, 64, 93, 176, 232, 576, 697, 988, 1160, 2048, 2325, 2952, 3304, 4992, 5489, 6580, 7176, 9920, 10701, 12384, 13288, 17344, 18473, 20876, 22152, 27776, 29317, 32568, 34280, 41728, 43745, 47972, 50184, 59712, 62269, 67600, 70376, 82240, 85401, 91964, 95368, 109824, 113653, 121576, 125672, 142976, 147537, 156948, 161800, 182208, 187565, 198592, 204264, 228032, 234249, 247020, 253576, 280960, 288101, 302744, 310248, 341504, 349633, 366276, 374792, 410176, 419357, 438128, 447720, 487488, 497785, 518812, 529544, 573952, 585429, 608840, 620776, 670080, 682801, 708724, 721928, 776384, 790413, 818976, 833512, 893376, 908777, 940108, 956040, 1021568, 1038405, 1072632, 1090024, 1161472, 1179809, 1217060, 1235976, 1313600, 1333501, 1373904, 1394408, 1478464, 1499993, 1543676, 1565832, 1656576, 1679797, 1726888, 1750760, 1848448, 1873425, 1924052, 1949704, 2054592, 2081389, 2135680, 2163176, 2275520, 2304201, 2362284, 2391688, 2511744, 2542373, 2604376, 2635752, 2763776, 2796417, 2862468, 2895880, 3032128, 3066845, 3137072, 3172584, 3317312, 3354169, 3428700, 3466376, 3619840, 3658901, 3737864, 3777768, 3940224, 3981553, 4065076, 4107272, 4278976, 4322637, 4410848, 4455400, 4636608, 4682665, 4775692, 4822664, 5013632, 5062149, 5160120, 5209576, 5410560, 5461601, 5564644, 5616648, 5827904, 5881533, 5989776, 6044392, 6266176, 6322457, 6436028, 6493320, 6725888, 6784885, 6903912, 6963944, 7207552, 7269329, 7393940, 7456776, 7711680, 7776301, 7906624, 7972328, 8238784, 8306313, 8442476, 8511112, 8789376, 8859877, 9002008, 9073640, 9363968, 9437505, 9585732, 9660424, 9963072, 10039709, 10194160, 10271976, 10587200, 10667001, 10827804, 10908808, 11236864, 11319893, 11487176, 11571432, 11912576, 11998897, 12172788, 12260360, 12614848, 12704525, 12885152, 12976104, 13344192, 13437289, 13624780, 13719176, 14101120, 14197701, 14392184, 14490088, 14886144, 14986273, 15187876, 15289352, 15699776, 15803517, 16012368, 16117480, 16542528, 16649945, 16866172, 16974984, 17414912, 17526069, 17749800, 17862376, 18317440, 18432401, 18663764, 18780168, 19250624, 19369453, 19608576, 19728872, 20214976, 20337737, 20584748, 20709000, 21211008, 21337765, 21592792, 21721064, 22239232, 22370049, 22633220, 22765576, 23300160, 23435101, 23706544, 23843048, 24394304, 24533433, 24813276, 24953992, 25522176, 25665557, 25953928, 26098920, 26684288, 26831985, 27129012, 27278344, 27881152, 28033229, 28339040, 28492776, 29113280, 29269801.

Some larger examples :