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Achilles n.  and  amenable n.

First numbers that belong to both sets:

72, 108, 200, 288, 392, 432, 500, 648, 800, 864, 968, 972, 1125, 1152, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3200, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6125, 6272, 6728, 6912, 7200, 7688, 8712, 8748, 8788, 9000, 9248, 9800, 10125, 10368, 10584, 10800, 10952, 10976, 11552, 12168, 12348, 12500, 12800, 13068, 13448, 13500, 14112, 14792, 15125, 15488, 15552, 16000, 16200, 16928, 17496, 17672, 18000, 18252, 18432, 19208, 19652, 19773, 20000, 20808, 21125, 21168, 21296, 21600, 21632, 22472, 23328, 24200, 24300, 24500, 24696, 25000, 25088, 25992, 26136, 26912, 27436, 27648, 27848, 28125, 28800, 29768, 30752, 31104, 31212, 31752, 32000, 33800, 34300, 34848, 34992, 35152, 35912, 36000, 36125, 36504, 36992, 37044, 38088, 38988, 39200, 40328, 40500, 41472, 42336, 42592, 42632, 43200, 43808, 43904, 44217, 45000, 45125, 46208, 47432, 47628, 47916, 48600, 48668, 48672, 49000, 49392, 49928, 50000, 51200, 52272, 52488, 53792, 54000, 54925, 55112, 55125, 55296, 56448, 57132, 57800, 59168, 60500, 60552, 61952, 62208, 62424, 63368, 64800, 66125, 66248, 67228, 67500, 67712, 68600, 69192, 69984, 70304, 70688, 72000, 72200, 73008, 73728, 75272, 76832, 77976, 78408, 78608, 78732, 79092, 80000, 81000, 81608, 83232, 83349, 84500, 84672, 84872, 86400, 86528, 87808, 88200, 89888, 90828, 91592, 93312, 95048, 95256, 95832, 96800, 97200, 97556, 98000, 98568, 98784, 100352, 102152, 103788, 103968, 104544, 105125, 105800, 107648, 107653, 108000, 109512, 109744, 111132, 111392, 112500, 113288, 114264, 115200, 117128, 117612, 119072, 119164, 120125, 121000, 121032, 121500, 122825, 123008, 124416, 124848, 127008, 128000, 129032, 132300, 133100, 133128, 134456, 135000, 135200, 136125, 137200, 137288, 139392, 139968, 141512, 143648, 143748, 144000, 144500, 145800, 146016, 147852, 147968, 148176, 150152, 152352, 153125, 154568, 155952, 156800, 157216, 158184, 159048, 161312, 162000, 163592, 164268.

Some larger examples :