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amenable n.  and  octagonal n.

First numbers that belong to both sets:

1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461, 5720, 5985, 6256, 6533, 6816, 7105, 7400, 7701, 8008, 8321, 8640, 8965, 9296, 9633, 9976, 10325, 10680, 11041, 11408, 11781, 12160, 12545, 12936, 13333, 13736, 14145, 14560, 14981, 15408, 15841, 16280, 16725, 17176, 17633, 18096, 18565, 19040, 19521, 20008, 20501, 21000, 21505, 22016, 22533, 23056, 23585, 24120, 24661, 25208, 25761, 26320, 26885, 27456, 28033, 28616, 29205, 29800, 30401, 31008, 31621, 32240, 32865, 33496, 34133, 34776, 35425, 36080, 36741, 37408, 38081, 38760, 39445, 40136, 40833, 41536, 42245, 42960, 43681, 44408, 45141, 45880, 46625, 47376, 48133, 48896, 49665, 50440, 51221, 52008, 52801, 53600, 54405, 55216, 56033, 56856, 57685, 58520, 59361, 60208, 61061, 61920, 62785, 63656, 64533, 65416, 66305, 67200, 68101, 69008, 69921, 70840, 71765, 72696, 73633, 74576, 75525, 76480, 77441, 78408, 79381, 80360, 81345, 82336, 83333, 84336, 85345, 86360, 87381, 88408, 89441, 90480, 91525, 92576, 93633, 94696, 95765, 96840, 97921, 99008, 100101, 101200, 102305, 103416, 104533, 105656, 106785, 107920, 109061, 110208, 111361, 112520, 113685, 114856, 116033, 117216, 118405, 119600, 120801, 122008, 123221, 124440, 125665, 126896, 128133, 129376, 130625, 131880, 133141, 134408, 135681, 136960, 138245, 139536, 140833, 142136, 143445, 144760, 146081, 147408, 148741, 150080, 151425, 152776, 154133, 155496, 156865, 158240, 159621, 161008, 162401, 163800, 165205, 166616, 168033, 169456, 170885, 172320, 173761, 175208, 176661, 178120, 179585, 181056, 182533, 184016, 185505, 187000, 188501, 190008, 191521, 193040, 194565, 196096, 197633, 199176, 200725, 202280, 203841, 205408, 206981, 208560, 210145, 211736, 213333, 214936, 216545, 218160, 219781, 221408, 223041, 224680, 226325, 227976, 229633, 231296, 232965, 234640.

Some larger examples :