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amenable n.  and  taxicab n.

First numbers that belong to both sets:

1729, 4104, 13832, 32832, 39312, 40033, 64232, 65728, 110656, 110808, 149389, 165464, 171288, 195841, 216125, 262656, 314496, 320264, 373464, 402597, 439101, 443889, 513000, 513856, 525824, 558441, 704977, 805688, 885248, 886464, 920673, 955016, 994688, 1009736, 1016496, 1061424, 1075032, 1092728, 1195112, 1260441, 1323712, 1331064, 1370304, 1407672, 1533357, 1566728, 1609272, 1728216, 1729000, 1734264, 1774656, 1845649, 2101248, 2515968, 2562112, 2622104, 2864288, 2987712, 2991816, 3220776, 3242197, 3375001, 3375008, 3511872, 3512808, 3551112, 3587409, 3628233, 3798613, 3813992, 4104000, 4110848, 4123000, 4174281, 4206592, 4467528, 4505949, 4511808, 4607064, 4624776, 4673088, 4744376, 4914000, 4931101, 5004125, 5135104, 5318677, 5462424, 5472152, 5639816, 5832729, 6013000, 6017193, 6230504, 6445504, 6517000, 6742008, 7081984, 7091712, 7094269, 7365384, 7620661, 7640128, 7872536, 7882245, 7957504, 8029000, 8077888, 8131968, 8216000, 8275176, 8491392, 8494577, 8587000, 8600256, 8647128, 8741824, 8849601, 9016488, 9261729, 9443761, 9560896, 10036845, 10083528, 10202696, 10424232, 10469816, 10589696, 10648512, 10811017, 10821896, 10962432, 11261376, 11478584, 11548089, 11557000, 12266856, 12533824, 12650337, 12753160, 12874176, 13484016, 13490533, 13623913, 13825728, 13832000, 13851000, 13874112, 13915125, 14197248, 14765192, 14978061, 15187041, 15201368, 15398936, 15682680, 16012269, 16387128, 16387189, 16809984, 17999072, 18410392, 18468513, 18617625, 18673625, 19346168, 20072017, 20127744, 20162952, 20496896, 20683000, 20976832, 21125384, 21411000, 21531608, 21753576, 22031576, 22426369, 22543417, 22544704, 22754277, 22914304, 23901696, 23934528, 24480125, 24666560, 24897817, 25164405, 25766208, 25785432, 25937576, 26569809, 26856576, 27000008, 27000064, 27015625, 27262872, 27445392, 27529073, 28094976, 28102464, 28149336, 28408896, 28658448, 28699272, 28703269, 28981125, 29025864, 29184057, 29503656, 29791125, 30388904, 30511936, 31256000, 32268024, 32530680, 32832000, 32886784, 32984000, 33108264, 33394248, 33652736, 34743312, 35740224, 35938728, 36047592, 36050904, 36094464, 36856512, 36998208, 37384704, 37498032, 37955008, 38007144, 39312000, 39448808, 40033000, 40134584, 41080832, 41258737, 41301953, 41343640, 41436873, 41511232, 42154840, 42168581, 42301656, 42549416, 43450344, 43570872, 43699392, 43777216, 43852536, 43946000, 43997625, 45118528, 46091997, 46179224, 46394712, 46661832, 46683000, 46825128, 47915712, 48104000, 48137544, 48469976, 49844032, 49849856, 49933368, 50324625, 51278929, 51479064, 51564032, 52136000, 52324272, 52739288, 53226297, 53936064, 54814509, 54887625.

Some larger examples :